Continuous function

<h2 id="history">History</h2>
A form of the <a href="/facts/Limit_of_a_function/HFbQ69e5">epsilon–delta definition of continuity</a> was first given by <a href="/facts/Bernard_Bolzano/BUbRVY4p">Bernard Bolzano</a> in 1817. <a href="/facts/Augustin-Louis_Cauchy/aoTjgj6j">Augustin-Louis Cauchy</a> defined continuity of 
 
 
 
 y
 =
 f
 (
 x
 )
 
 
 {\displaystyle y=f(x)}
 
 as follows: an infinitely small increment 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
 of the independent variable x always produces an infinitely small change 
 
 
 
 f
 (
 x
 +
 α
 )
 −
 f
 (
 x
 )
 
 
 {\displaystyle f(x+\alpha )-f(x)}
 
 of the dependent variable y (see e.g. <a href="/facts/Cours_d%27Analyse/IUUvmV23">Cours d'Analyse</a>, p. 34). Cauchy defined infinitely small quantities in terms of variable quantities, and his definition of continuity closely parallels the infinitesimal definition used today (see <a href="/facts/Microcontinuity/h3fxbJHs">microcontinuity</a>). The formal definition and the distinction between pointwise continuity and <a href="/facts/Uniform_continuity/hjvw7tII">uniform continuity</a> were first given by Bolzano in the 1830s, but the work wasn't published until the 1930s. Like Bolzano,<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> <a href="/facts/Karl_Weierstrass/4U8vdQzL">Karl Weierstrass</a><a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> denied continuity of a function at a point c unless it was defined at and on both sides of c, but <a href="/facts/%C3%89douard_Goursat/kMndmd26">Édouard Goursat</a><a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> allowed the function to be defined only at and on one side of c, and <a href="/facts/Camille_Jordan/gmLuEJ4e">Camille Jordan</a><a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> allowed it even if the function was defined only at c. All three of those nonequivalent definitions of pointwise continuity are still in use.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> <a href="/facts/Eduard_Heine/FY8enC39">Eduard Heine</a> provided the first published definition of uniform continuity in 1872, but based these ideas on lectures given by <a href="/facts/Peter_Gustav_Lejeune_Dirichlet/b5UIU7Uy">Peter Gustav Lejeune Dirichlet</a> in 1854.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

<h2 id="real-functions">Real functions</h2>
<h3>Definition</h3>

A <a href="/facts/Real_function/bbNG3DnN">real function</a> that is a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> from <a href="/facts/Real_number/R02gw5Pb">real numbers</a> to real numbers can be represented by a <a href="/facts/Graph_of_a_function/htYX0BGX">graph</a> in the <a href="/facts/Cartesian_coordinate_system/lkTqvMmB">Cartesian plane</a>; such a function is continuous if, roughly speaking, the graph is a single unbroken <a href="/facts/Curve/xBQ0pTBW">curve</a> whose <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> is the entire real line. A more mathematically rigorous definition is given below.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>
Continuity of real functions is usually defined in terms of <a href="/facts/Limit_(mathematics)/j8JBhj4e">limits</a>. A function f with variable x is continuous at the <a href="/facts/Real_number/R02gw5Pb">real number</a> c, if the limit of 
 
 
 
 f
 (
 x
 )
 ,
 
 
 {\displaystyle f(x),}
 
 as x tends to c, is equal to 
 
 
 
 f
 (
 c
 )
 .
 
 
 {\displaystyle f(c).}

There are several different definitions of the (global) continuity of a function, which depend on the nature of its <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a>. 
A function is continuous on an <a href="/facts/Open_interval/e9se3vTe">open interval</a> if the interval is contained in the function's domain and the function is continuous at every interval point. A function that is continuous on the interval 
 
 
 
 (
 −
 ∞
 ,
 +
 ∞
 )
 
 
 {\displaystyle (-\infty ,+\infty )}
 
 (the whole <a href="/facts/Real_line/6eq42xX5">real line</a>) is often called simply a continuous function; one also says that such a function is continuous everywhere. For example, all <a href="/facts/Polynomial_function/Lzak8VVx">polynomial functions</a> are continuous everywhere.
A function is continuous on a <a href="/facts/Semi-open_interval/e9se3vTe">semi-open</a> or a <a href="/facts/Closed_interval/e9se3vTe">closed</a> interval; if the interval is contained in the domain of the function, the function is continuous at every interior point of the interval, and the value of the function at each endpoint that belongs to the interval is the limit of the values of the function when the variable tends to the endpoint from the interior of the interval. For example, the function 
 
 
 
 f
 (
 x
 )
 =
 
 
 x
 
 
 
 
 {\displaystyle f(x)={\sqrt {x}}}
 
 is continuous on its whole domain, which is the closed interval 
 
 
 
 [
 0
 ,
 +
 ∞
 )
 .
 
 
 {\displaystyle [0,+\infty ).}

Many commonly encountered functions are <a href="/facts/Partial_function/0vJMw0Nm">partial functions</a> that have a domain formed by all real numbers, except some <a href="/facts/Isolated_point/d3EeWxuD">isolated points</a>. Examples include the <a href="/facts/Reciprocal_function/yczdYyCw">reciprocal function</a> 
 
 
 
 x
 ↦
 
 
 1
 x
 
 
 
 
 {\textstyle x\mapsto {\frac {1}{x}}}
 
 and the <a href="/facts/Tangent_function/czej7ur0">tangent function</a> 
 
 
 
 x
 ↦
 tan
 ⁡
 x
 .
 
 
 {\displaystyle x\mapsto \tan x.}
 
 When they are continuous on their domain, one says, in some contexts, that they are continuous, although they are not continuous everywhere. In other contexts, mainly when one is interested in their behavior near the exceptional points, one says they are discontinuous.
A partial function is discontinuous at a point if the point belongs to the <a href="/facts/Topological_closure/9x2wlgUW">topological closure</a> of its domain, and either the point does not belong to the domain of the function or the function is not continuous at the point. For example, the functions 
 
 
 
 x
 ↦
 
 
 1
 x
 
 
 
 
 {\textstyle x\mapsto {\frac {1}{x}}}
 
 and 
 
 
 
 x
 ↦
 sin
 ⁡
 (
 
 
 1
 x
 
 
 )
 
 
 {\textstyle x\mapsto \sin({\frac {1}{x}})}
 
 are discontinuous at 0, and remain discontinuous whichever value is chosen for defining them at 0. A point where a function is discontinuous is called a discontinuity.
Using mathematical notation, several ways exist to define continuous functions in the three senses mentioned above.
Let 
 
 
 
 f
 :
 D
 →
 
 R
 
 
 
 {\displaystyle f:D\to \mathbb {R} }
 
 be a function defined on a <a href="/facts/Subset/SJGERmbA">subset</a> 
 
 
 
 D
 
 
 {\displaystyle D}
 
 of the set 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
 of real numbers.
This subset 
 
 
 
 D
 
 
 {\displaystyle D}
 
 is the domain of f. Some possible choices include 

<ul><li>
 
 
 
 D
 =
 
 R
 
 
 
 {\displaystyle D=\mathbb {R} }
 
: i.e., 
 
 
 
 D
 
 
 {\displaystyle D}
 
 is the whole set of real numbers. or, for a and b real numbers,</li>
<li>
 
 
 
 D
 =
 [
 a
 ,
 b
 ]
 =
 {
 x
 ∈
 
 R
 
 ∣
 a
 ≤
 x
 ≤
 b
 }
 
 
 {\displaystyle D=[a,b]=\{x\in \mathbb {R} \mid a\leq x\leq b\}}
 
: 
 
 
 
 D
 
 
 {\displaystyle D}
 
 is a <a href="/facts/Closed_interval/e9se3vTe">closed interval</a>, or</li>
<li>
 
 
 
 D
 =
 (
 a
 ,
 b
 )
 =
 {
 x
 ∈
 
 R
 
 ∣
 a
 <
 x
 open interval</a>.</li></ul>
In the case of the domain 
 
 
 
 D
 
 
 {\displaystyle D}
 
 being defined as an open interval, 
 
 
 
 a
 
 
 {\displaystyle a}
 
 and 
 
 
 
 b
 
 
 {\displaystyle b}
 
 do not belong to 
 
 
 
 D
 
 
 {\displaystyle D}
 
, and the values of 
 
 
 
 f
 (
 a
 )
 
 
 {\displaystyle f(a)}
 
 and 
 
 
 
 f
 (
 b
 )
 
 
 {\displaystyle f(b)}
 
 do not matter for continuity on 
 
 
 
 D
 
 
 {\displaystyle D}
 
.

<h4>Definition in terms of limits of functions</h4>
The function f is continuous at some point c of its domain if the <a href="/facts/Limit_of_a_function/HFbQ69e5">limit</a> of 
 
 
 
 f
 (
 x
 )
 ,
 
 
 {\displaystyle f(x),}
 
 as x approaches c through the domain of f, exists and is equal to 
 
 
 
 f
 (
 c
 )
 .
 
 
 {\displaystyle f(c).}
 
<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> In mathematical notation, this is written as

lim
          
            x
            →
            c
          
        
        
          f
          (
          x
          )
        
        =
        f
        (
        c
        )
        .
      
    
    {\displaystyle \lim _{x\to c}{f(x)}=f(c).}

In detail this means three conditions: first, f has to be defined at c (guaranteed by the requirement that c is in the domain of f). Second, the limit of that equation has to exist. Third, the value of this limit must equal 
 
 
 
 f
 (
 c
 )
 .
 
 
 {\displaystyle f(c).}

(Here, we have assumed that the domain of f does not have any <a href="/facts/Isolated_point/d3EeWxuD">isolated points</a>.)

<h4>Definition in terms of neighborhoods</h4>
A <a href="/facts/Neighborhood_(mathematics)/XHKgrpkp">neighborhood</a> of a point c is a set that contains, at least, all points within some fixed distance of c. Intuitively, a function is continuous at a point c if the range of f over the neighborhood of c shrinks to a single point 
 
 
 
 f
 (
 c
 )
 
 
 {\displaystyle f(c)}
 
 as the width of the neighborhood around c shrinks to zero. More precisely, a function f is continuous at a point c of its domain if, for any neighborhood 
 
 
 
 
 N
 
 1
 
 
 (
 f
 (
 c
 )
 )
 
 
 {\displaystyle N_{1}(f(c))}
 
 there is a neighborhood 
 
 
 
 
 N
 
 2
 
 
 (
 c
 )
 
 
 {\displaystyle N_{2}(c)}
 
 in its domain such that 
 
 
 
 f
 (
 x
 )
 ∈
 
 N
 
 1
 
 
 (
 f
 (
 c
 )
 )
 
 
 {\displaystyle f(x)\in N_{1}(f(c))}
 
 whenever 
 
 
 
 x
 ∈
 
 N
 
 2
 
 
 (
 c
 )
 .
 
 
 {\displaystyle x\in N_{2}(c).}

As neighborhoods are defined in any <a href="/facts/Topological_space/v2ut7CbK">topological space</a>, this definition of a continuous function applies not only for real functions but also when the domain and the <a href="/facts/Codomain/MGc43nwQ">codomain</a> are <a href="/facts/Topological_space/v2ut7CbK">topological spaces</a> and is thus the most general definition. It follows that a function is automatically continuous at every <a href="/facts/Isolated_point/d3EeWxuD">isolated point</a> of its domain. For example, every real-valued function on the integers is continuous.

<h4>Definition in terms of limits of sequences</h4>

One can instead require that for any <a href="/facts/Sequence_(mathematics)/gV2S4uqp">sequence</a> 
 
 
 
 (
 
 x
 
 n
 
 
 
 )
 
 n
 ∈
 
 N
 
 
 
 
 
 {\displaystyle (x_{n})_{n\in \mathbb {N} }}
 
 of points in the domain which <a href="/facts/Convergent_sequence/Ktge2RwL">converges</a> to c, the corresponding sequence 
 
 
 
 
 
 (
 
 f
 (
 
 x
 
 n
 
 
 )
 
 )
 
 
 n
 ∈
 
 N
 
 
 
 
 
 {\displaystyle \left(f(x_{n})\right)_{n\in \mathbb {N} }}
 
 converges to 
 
 
 
 f
 (
 c
 )
 .
 
 
 {\displaystyle f(c).}
 
 In mathematical notation, 
 
 
 
 ∀
 (
 
 x
 
 n
 
 
 
 )
 
 n
 ∈
 
 N
 
 
 
 ⊂
 D
 :
 
 lim
 
 n
 →
 ∞
 
 
 
 x
 
 n
 
 
 =
 c
 ⇒
 
 lim
 
 n
 →
 ∞
 
 
 f
 (
 
 x
 
 n
 
 
 )
 =
 f
 (
 c
 )
 
 .
 
 
 {\displaystyle \forall (x_{n})_{n\in \mathbb {N} }\subset D:\lim _{n\to \infty }x_{n}=c\Rightarrow \lim _{n\to \infty }f(x_{n})=f(c)\,.}

<h4>Weierstrass and Jordan definitions (epsilon–delta) of continuous functions</h4>

Explicitly including the definition of the limit of a function, we obtain a self-contained definition: Given a function 
 
 
 
 f
 :
 D
 →
 
 R
 
 
 
 {\displaystyle f:D\to \mathbb {R} }
 
 as above and an element 
 
 
 
 
 x
 
 0
 
 
 
 
 {\displaystyle x_{0}}
 
 of the domain 
 
 
 
 D
 
 
 {\displaystyle D}
 
, 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is said to be continuous at the point 
 
 
 
 
 x
 
 0
 
 
 
 
 {\displaystyle x_{0}}
 
 when the following holds: For any positive real number 
 
 
 
 ε
 >
 0
 ,
 
 
 {\displaystyle \varepsilon >0,}
 
 however small, there exists some positive real number 
 
 
 
 δ
 >
 0
 
 
 {\displaystyle \delta >0}
 
 such that for all 
 
 
 
 x
 
 
 {\displaystyle x}
 
 in the domain of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 with 
 
 
 
 
 x
 
 0
 
 
 −
 δ
 <
 x
 <
 
 x
 
 0
 
 
 +
 δ
 ,
 
 
 {\displaystyle x_{0}-\delta <x<x_{0}+\delta ,}
 
 the value of 
 
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(x)}
 
 satisfies

f
 
 (
 
 x
 
 0
 
 
 )
 
 −
 ε
 <
 f
 (
 x
 )
 <
 f
 (
 
 x
 
 0
 
 
 )
 +
 ε
 .
 
 
 {\displaystyle f\left(x_{0}\right)-\varepsilon <f(x)<f(x_{0})+\varepsilon .}

Alternatively written, continuity of 
 
 
 
 f
 :
 D
 →
 
 R
 
 
 
 {\displaystyle f:D\to \mathbb {R} }
 
 at 
 
 
 
 
 x
 
 0
 
 
 ∈
 D
 
 
 {\displaystyle x_{0}\in D}
 
 means that for every 
 
 
 
 ε
 >
 0
 ,
 
 
 {\displaystyle \varepsilon >0,}
 
 there exists a 
 
 
 
 δ
 >
 0
 
 
 {\displaystyle \delta >0}
 
 such that for all 
 
 
 
 x
 ∈
 D
 
 
 {\displaystyle x\in D}
 
:

|
 
 x
 −
 
 x
 
 0
 
 
 
 |
 
 <
 δ
  
  
 
  implies 
 
  
  
 
 |
 
 f
 (
 x
 )
 −
 f
 (
 
 x
 
 0
 
 
 )
 
 |
 
 <
 ε
 .
 
 
 {\displaystyle \left|x-x_{0}\right|<\delta ~~{\text{ implies }}~~|f(x)-f(x_{0})|<\varepsilon .}

More intuitively, we can say that if we want to get all the 
 
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(x)}
 
 values to stay in some small <a href="/facts/Topological_neighborhood/XHKgrpkp">neighborhood</a> around 
 
 
 
 f
 
 (
 
 x
 
 0
 
 
 )
 
 ,
 
 
 {\displaystyle f\left(x_{0}\right),}
 
 we need to choose a small enough neighborhood for the 
 
 
 
 x
 
 
 {\displaystyle x}
 
 values around 
 
 
 
 
 x
 
 0
 
 
 .
 
 
 {\displaystyle x_{0}.}
 
 If we can do that no matter how small the 
 
 
 
 f
 (
 
 x
 
 0
 
 
 )
 
 
 {\displaystyle f(x_{0})}
 
 neighborhood is, then 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is continuous at 
 
 
 
 
 x
 
 0
 
 
 .
 
 
 {\displaystyle x_{0}.}

In modern terms, this is generalized by the definition of continuity of a function with respect to a <a href="/facts/Basis_(topology)/8EDSDVPD">basis for the topology</a>, here the <a href="/facts/Metric_topology/gnBBRXtc">metric topology</a>.
Weierstrass had required that the interval 
 
 
 
 
 x
 
 0
 
 
 −
 δ
 <
 x
 <
 
 x
 
 0
 
 
 +
 δ
 
 
 {\displaystyle x_{0}-\delta <x<x_{0}+\delta }
 
 be entirely within the domain 
 
 
 
 D
 
 
 {\displaystyle D}
 
, but Jordan removed that restriction.

<h4>Definition in terms of control of the remainder</h4>
In proofs and numerical analysis, we often need to know how fast limits are converging, or in other words, control of the remainder. We can formalize this to a definition of continuity. 
A function 
 
 
 
 C
 :
 [
 0
 ,
 ∞
 )
 →
 [
 0
 ,
 ∞
 ]
 
 
 {\displaystyle C:[0,\infty )\to [0,\infty ]}
 
 is called a control function if

<ul><li>C is non-decreasing</li>
<li>
 
 
 
 
 inf
 
 δ
 >
 0
 
 
 C
 (
 δ
 )
 =
 0
 
 
 {\displaystyle \inf _{\delta >0}C(\delta )=0}
 
</li></ul>
A function 
 
 
 
 f
 :
 D
 →
 R
 
 
 {\displaystyle f:D\to R}
 
 is C-continuous at 
 
 
 
 
 x
 
 0
 
 
 
 
 {\displaystyle x_{0}}
 
 if there exists such a neighbourhood 
 
 
 
 N
 (
 
 x
 
 0
 
 
 )
 
 
 {\textstyle N(x_{0})}
 
 that

|
        
        f
        (
        x
        )
        −
        f
        (
        
          x
          
            0
          
        
        )
        
          |
        
        ≤
        C
        
          (
          
            |
            
              x
              −
              
                x
                
                  0
                
              
            
            |
          
          )
        
        
           for all 
        
        x
        ∈
        D
        ∩
        N
        (
        
          x
          
            0
          
        
        )
      
    
    {\displaystyle |f(x)-f(x_{0})|\leq C\left(\left|x-x_{0}\right|\right){\text{ for all }}x\in D\cap N(x_{0})}

A function is continuous in 
 
 
 
 
 x
 
 0
 
 
 
 
 {\displaystyle x_{0}}
 
 if it is C-continuous for some control function C.
This approach leads naturally to refining the notion of continuity by restricting the set of admissible control functions. For a given set of control functions 
 
 
 
 
 
 C
 
 
 
 
 {\displaystyle {\mathcal {C}}}
 
 a function is 
 
 
 
 
 
 C
 
 
 
 
 {\displaystyle {\mathcal {C}}}
 
-continuous if it is 
 
 
 
 C
 
 
 {\displaystyle C}
 
-continuous for some 
 
 
 
 C
 ∈
 
 
 C
 
 
 .
 
 
 {\displaystyle C\in {\mathcal {C}}.}
 
 For example, the <a href="/facts/Lipschitz_continuity/Hw20EPEU">Lipschitz</a> and <a href="/facts/H%C3%B6lder_continuous_function/CI0yp6kD">Hölder continuous functions</a> of exponent α below are defined by the set of control functions

C
            
          
          
            
              L
              i
              p
              s
              c
              h
              i
              t
              z
            
          
        
        =
        {
        C
        :
        C
        (
        δ
        )
        =
        K
        
          |
        
        δ
        
          |
        
        ,
         
        K
        >
        0
        }
      
    
    {\displaystyle {\mathcal {C}}_{\mathrm {Lipschitz} }=\{C:C(\delta )=K|\delta |,\ K>0\}}
  
 
respectively

C
            
          
          
            
              Hölder
            
            −
            α
          
        
        =
        {
        C
        :
        C
        (
        δ
        )
        =
        K
        
          |
        
        δ
        
          
            |
          
          
            α
          
        
        ,
         
        K
        >
        0
        }
        .
      
    
    {\displaystyle {\mathcal {C}}_{{\text{Hölder}}-\alpha }=\{C:C(\delta )=K|\delta |^{\alpha },\ K>0\}.}

<h4>Definition using oscillation</h4>

Continuity can also be defined in terms of <a href="/facts/Oscillation_(mathematics)/JHMgNIhG">oscillation</a>: a function f is continuous at a point 
 
 
 
 
 x
 
 0
 
 
 
 
 {\displaystyle x_{0}}
 
 if and only if its oscillation at that point is zero;<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> in symbols, 
 
 
 
 
 ω
 
 f
 
 
 (
 
 x
 
 0
 
 
 )
 =
 0.
 
 
 {\displaystyle \omega _{f}(x_{0})=0.}
 
 A benefit of this definition is that it quantifies discontinuity: the oscillation gives how much the function is discontinuous at a point.
This definition is helpful in <a href="/facts/Descriptive_set_theory/ek0tvUXb">descriptive set theory</a> to study the set of discontinuities and continuous points – the continuous points are the intersection of the sets where the oscillation is less than 
 
 
 
 ε
 
 
 {\displaystyle \varepsilon }
 
 (hence a <a href="/facts/G-delta_set/Erg1BCFk">
 
 
 
 
 G
 
 δ
 
 
 
 
 {\displaystyle G_{\delta }}
 
 set</a>) – and gives a rapid proof of one direction of the <a href="/facts/Lebesgue_integrability_condition/vWGrPhVh">Lebesgue integrability condition</a>.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>
The oscillation is equivalent to the 
 
 
 
 ε
 −
 δ
 
 
 {\displaystyle \varepsilon -\delta }
 
 definition by a simple re-arrangement and by using a limit (<a href="/facts/Lim_sup/QeAhlygs">lim sup</a>, <a href="/facts/Lim_inf/QeAhlygs">lim inf</a>) to define oscillation: if (at a given point) for a given 
 
 
 
 
 ε
 
 0
 
 
 
 
 {\displaystyle \varepsilon _{0}}
 
 there is no 
 
 
 
 δ
 
 
 {\displaystyle \delta }
 
 that satisfies the 
 
 
 
 ε
 −
 δ
 
 
 {\displaystyle \varepsilon -\delta }
 
 definition, then the oscillation is at least 
 
 
 
 
 ε
 
 0
 
 
 ,
 
 
 {\displaystyle \varepsilon _{0},}
 
 and conversely if for every 
 
 
 
 ε
 
 
 {\displaystyle \varepsilon }
 
 there is a desired 
 
 
 
 δ
 ,
 
 
 {\displaystyle \delta ,}
 
 the oscillation is 0. The oscillation definition can be naturally generalized to maps from a topological space to a <a href="/facts/Metric_space/gnBBRXtc">metric space</a>.

<h4>Definition using the hyperreals</h4>
<a href="/facts/Cauchy/aoTjgj6j">Cauchy</a> defined the continuity of a function in the following intuitive terms: an <a href="/facts/Infinitesimal/JKcTMIgD">infinitesimal</a> change in the independent variable corresponds to an infinitesimal change of the dependent variable (see Cours d'analyse, page 34). <a href="/facts/Non-standard_analysis/lEpNpWMB">Non-standard analysis</a> is a way of making this mathematically rigorous. The real line is augmented by adding infinite and infinitesimal numbers to form the <a href="/facts/Hyperreal_numbers/Kvf09qpo">hyperreal numbers</a>. In nonstandard analysis, continuity can be defined as follows.

A real-valued function f is continuous at x if its natural extension to the hyperreals has the property that for all infinitesimal dx, 
 
 
 
 f
 (
 x
 +
 d
 x
 )
 −
 f
 (
 x
 )
 
 
 {\displaystyle f(x+dx)-f(x)}
 
 is infinitesimal<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>
(see <a href="/facts/Microcontinuity/h3fxbJHs">microcontinuity</a>). In other words, an infinitesimal increment of the independent variable always produces an infinitesimal change of the dependent variable, giving a modern expression to <a href="/facts/Augustin-Louis_Cauchy/aoTjgj6j">Augustin-Louis Cauchy</a>'s definition of continuity.

<h3>Construction of continuous functions</h3>

Checking the continuity of a given function can be simplified by checking one of the above defining properties for the building blocks of the given function. It is straightforward to show that the sum of two functions, continuous on some domain, is also continuous on this domain. Given

f
        ,
        g
        :
        D
        →
        
          R
        
        ,
      
    
    {\displaystyle f,g\colon D\to \mathbb {R} ,}

then the sum of continuous functions

s
        =
        f
        +
        g
      
    
    {\displaystyle s=f+g}
  
 
(defined by 
  
    
      
        s
        (
        x
        )
        =
        f
        (
        x
        )
        +
        g
        (
        x
        )
      
    
    {\displaystyle s(x)=f(x)+g(x)}
  
 for all 
  
    
      
        x
        ∈
        D
      
    
    {\displaystyle x\in D}
  
) is continuous in 
  
    
      
        D
        .
      
    
    {\displaystyle D.}

The same holds for the product of continuous functions,

p
        =
        f
        ⋅
        g
      
    
    {\displaystyle p=f\cdot g}

(defined by 
  
    
      
        p
        (
        x
        )
        =
        f
        (
        x
        )
        ⋅
        g
        (
        x
        )
      
    
    {\displaystyle p(x)=f(x)\cdot g(x)}
  
 for all 
  
    
      
        x
        ∈
        D
      
    
    {\displaystyle x\in D}
  
)
is continuous in 
  
    
      
        D
        .
      
    
    {\displaystyle D.}

Combining the above preservations of continuity and the continuity of <a href="/facts/Constant_function/BmPx6dn7">constant functions</a> and of the <a href="/facts/Identity_function/9AtsP0LC">identity function</a> 
 
 
 
 I
 (
 x
 )
 =
 x
 
 
 {\displaystyle I(x)=x}
 
 on 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
, one arrives at the continuity of all <a href="/facts/Polynomial/Lzak8VVx">polynomial functions</a> on 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
, such as

f
        (
        x
        )
        =
        
          x
          
            3
          
        
        +
        
          x
          
            2
          
        
        −
        5
        x
        +
        3
      
    
    {\displaystyle f(x)=x^{3}+x^{2}-5x+3}

(pictured on the right).

In the same way, it can be shown that the reciprocal of a continuous function

r
        =
        1
        
          /
        
        f
      
    
    {\displaystyle r=1/f}

(defined by 
  
    
      
        r
        (
        x
        )
        =
        1
        
          /
        
        f
        (
        x
        )
      
    
    {\displaystyle r(x)=1/f(x)}
  
 for all 
  
    
      
        x
        ∈
        D
      
    
    {\displaystyle x\in D}
  
 such that 
  
    
      
        f
        (
        x
        )
        ≠
        0
      
    
    {\displaystyle f(x)\neq 0}
  
)
is continuous in 
  
    
      
        D
        ∖
        {
        x
        :
        f
        (
        x
        )
        =
        0
        }
        .
      
    
    {\displaystyle D\setminus \{x:f(x)=0\}.}

This implies that, excluding the roots of 
 
 
 
 g
 ,
 
 
 {\displaystyle g,}
 
 the quotient of continuous functions

q
        =
        f
        
          /
        
        g
      
    
    {\displaystyle q=f/g}

(defined by 
 
 
 
 q
 (
 x
 )
 =
 f
 (
 x
 )
 
 /
 
 g
 (
 x
 )
 
 
 {\displaystyle q(x)=f(x)/g(x)}
 
 for all 
 
 
 
 x
 ∈
 D
 
 
 {\displaystyle x\in D}
 
, such that 
 
 
 
 g
 (
 x
 )
 ≠
 0
 
 
 {\displaystyle g(x)\neq 0}
 
)
is also continuous on 
 
 
 
 D
 ∖
 {
 x
 :
 g
 (
 x
 )
 =
 0
 }
 
 
 {\displaystyle D\setminus \{x:g(x)=0\}}
 
.
For example, the function (pictured)

y
        (
        x
        )
        =
        
          
            
              2
              x
              −
              1
            
            
              x
              +
              2
            
          
        
      
    
    {\displaystyle y(x)={\frac {2x-1}{x+2}}}

is defined for all real numbers 
  
    
      
        x
        ≠
        −
        2
      
    
    {\displaystyle x\neq -2}
  
 and is continuous at every such point. Thus, it is a continuous function. The question of continuity at 
  
    
      
        x
        =
        −
        2
      
    
    {\displaystyle x=-2}
  
 does not arise since 
  
    
      
        x
        =
        −
        2
      
    
    {\displaystyle x=-2}
  
 is not in the domain of 
  
    
      
        y
        .
      
    
    {\displaystyle y.}
  
 There is no continuous function 
  
    
      
        F
        :
        
          R
        
        →
        
          R
        
      
    
    {\displaystyle F:\mathbb {R} \to \mathbb {R} }
  
 that agrees with 
  
    
      
        y
        (
        x
        )
      
    
    {\displaystyle y(x)}
  
 for all 
  
    
      
        x
        ≠
        −
        2.
      
    
    {\displaystyle x\neq -2.}

Since the function <a href="/facts/Sine/gQ6LXK8z">sine</a> is continuous on all reals, the <a href="/facts/Sinc_function/lSvDtHy0">sinc function</a> 
 
 
 
 G
 (
 x
 )
 =
 sin
 ⁡
 (
 x
 )
 
 /
 
 x
 ,
 
 
 {\displaystyle G(x)=\sin(x)/x,}
 
 is defined and continuous for all real 
 
 
 
 x
 ≠
 0.
 
 
 {\displaystyle x\neq 0.}
 
 However, unlike the previous example, G can be extended to a continuous function on all real numbers, by defining the value 
 
 
 
 G
 (
 0
 )
 
 
 {\displaystyle G(0)}
 
 to be 1, which is the limit of 
 
 
 
 G
 (
 x
 )
 ,
 
 
 {\displaystyle G(x),}
 
 when x approaches 0, i.e.,

G
        (
        0
        )
        =
        
          lim
          
            x
            →
            0
          
        
        
          
            
              sin
              ⁡
              x
            
            x
          
        
        =
        1.
      
    
    {\displaystyle G(0)=\lim _{x\to 0}{\frac {\sin x}{x}}=1.}

Thus, by setting

G
        (
        x
        )
        =
        
          
            {
            
              
                
                  
                    
                      
                        sin
                        ⁡
                        (
                        x
                        )
                      
                      x
                    
                  
                
                
                  
                     if 
                  
                  x
                  ≠
                  0
                
              
              
                
                  1
                
                
                  
                     if 
                  
                  x
                  =
                  0
                  ,
                
              
            
            
          
        
      
    
    {\displaystyle G(x)={\begin{cases}{\frac {\sin(x)}{x}}&{\text{ if }}x\neq 0\\1&{\text{ if }}x=0,\end{cases}}}

the sinc-function becomes a continuous function on all real numbers. The term <a href="/facts/Removable_singularity/6MPFDNha">removable singularity</a> is used in such cases when (re)defining values of a function to coincide with the appropriate limits make a function continuous at specific points.
A more involved construction of continuous functions is the <a href="/facts/Function_composition/r5Pvnmth">function composition</a>. Given two continuous functions

g
        :
        
          D
          
            g
          
        
        ⊆
        
          R
        
        →
        
          R
          
            g
          
        
        ⊆
        
          R
        
        
        
           and 
        
        
        f
        :
        
          D
          
            f
          
        
        ⊆
        
          R
        
        →
        
          R
          
            f
          
        
        ⊆
        
          D
          
            g
          
        
        ,
      
    
    {\displaystyle g:D_{g}\subseteq \mathbb {R} \to R_{g}\subseteq \mathbb {R} \quad {\text{ and }}\quad f:D_{f}\subseteq \mathbb {R} \to R_{f}\subseteq D_{g},}
  
 
their composition, denoted as

c
 =
 g
 ∘
 f
 :
 
 D
 
 f
 
 
 →
 
 R
 
 ,
 
 
 {\displaystyle c=g\circ f:D_{f}\to \mathbb {R} ,}
 
 and defined by 
 
 
 
 c
 (
 x
 )
 =
 g
 (
 f
 (
 x
 )
 )
 ,
 
 
 {\displaystyle c(x)=g(f(x)),}
 
 is continuous.
This construction allows stating, for example, that

e
          
            sin
            ⁡
            (
            ln
            ⁡
            x
            )
          
        
      
    
    {\displaystyle e^{\sin(\ln x)}}
  
 
is continuous for all 
  
    
      
        x
        >
        0.
      
    
    {\displaystyle x>0.}

<h3>Examples of discontinuous functions</h3>

An example of a discontinuous function is the <a href="/facts/Heaviside_step_function/bgsUtxgP">Heaviside step function</a> 
 
 
 
 H
 
 
 {\displaystyle H}
 
, defined by

H
 (
 x
 )
 =
 
 
 {
 
 
 
 1
 
 
 
  if 
 
 x
 ≥
 0
 
 
 
 
 0
 
 
 
  if 
 
 x
 <
 0
 
 
 
 
 
 
 
 
 {\displaystyle H(x)={\begin{cases}1&{\text{ if }}x\geq 0\\0&{\text{ if }}x<0\end{cases}}}

Pick for instance 
 
 
 
 ε
 =
 1
 
 /
 
 2
 
 
 {\displaystyle \varepsilon =1/2}
 
. Then there is no 
 
 
 
 δ
 
 
 {\displaystyle \delta }
 
-neighborhood around 
 
 
 
 x
 =
 0
 
 
 {\displaystyle x=0}
 
, i.e. no open interval 
 
 
 
 (
 −
 δ
 ,
 
 δ
 )
 
 
 {\displaystyle (-\delta ,\;\delta )}
 
 with 
 
 
 
 δ
 >
 0
 ,
 
 
 {\displaystyle \delta >0,}
 
 that will force all the 
 
 
 
 H
 (
 x
 )
 
 
 {\displaystyle H(x)}
 
 values to be within the 
 
 
 
 ε
 
 
 {\displaystyle \varepsilon }
 
-neighborhood of 
 
 
 
 H
 (
 0
 )
 
 
 {\displaystyle H(0)}
 
, i.e. within 
 
 
 
 (
 1
 
 /
 
 2
 ,
 
 3
 
 /
 
 2
 )
 
 
 {\displaystyle (1/2,\;3/2)}
 
. Intuitively, we can think of this type of discontinuity as a sudden <a href="/facts/Jump_discontinuity/TGLMsQom">jump</a> in function values.
Similarly, the <a href="/facts/Sign_function/VADy9zQq">signum</a> or sign function

sgn
 ⁡
 (
 x
 )
 =
 
 
 {
 
 
 
 
 
  
 1
 
 
 
  if 
 
 x
 >
 0
 
 
 
 
 
 
  
 0
 
 
 
  if 
 
 x
 =
 0
 
 
 
 
 −
 1
 
 
 
  if 
 
 x
 <
 0
 
 
 
 
 
 
 
 
 {\displaystyle \operatorname {sgn}(x)={\begin{cases}\;\;\ 1&{\text{ if }}x>0\\\;\;\ 0&{\text{ if }}x=0\\-1&{\text{ if }}x<0\end{cases}}}

is discontinuous at 
  
    
      
        x
        =
        0
      
    
    {\displaystyle x=0}
  
 but continuous everywhere else. Yet another example: the function

f
        (
        x
        )
        =
        
          
            {
            
              
                
                  sin
                  ⁡
                  
                    (
                    
                      x
                      
                        −
                        2
                      
                    
                    )
                  
                
                
                  
                     if 
                  
                  x
                  ≠
                  0
                
              
              
                
                  0
                
                
                  
                     if 
                  
                  x
                  =
                  0
                
              
            
            
          
        
      
    
    {\displaystyle f(x)={\begin{cases}\sin \left(x^{-2}\right)&{\text{ if }}x\neq 0\\0&{\text{ if }}x=0\end{cases}}}

is continuous everywhere apart from 
 
 
 
 x
 =
 0
 
 
 {\displaystyle x=0}
 
.

Besides plausible continuities and discontinuities like above, there are also functions with a behavior, often coined <a href="/facts/Pathological_(mathematics)/KmKHjJ1H">pathological</a>, for example, <a href="/facts/Thomae%27s_function/SjCt2ahN">Thomae's function</a>,

f
        (
        x
        )
        =
        
          
            {
            
              
                
                  1
                
                
                  
                     if 
                  
                  x
                  =
                  0
                
              
              
                
                  
                    
                      1
                      q
                    
                  
                
                
                  
                     if 
                  
                  x
                  =
                  
                    
                      p
                      q
                    
                  
                  
                    (in lowest terms) is a rational number
                  
                
              
              
                
                  0
                
                
                  
                     if 
                  
                  x
                  
                     is irrational
                  
                  .
                
              
            
            
          
        
      
    
    {\displaystyle f(x)={\begin{cases}1&{\text{ if }}x=0\\{\frac {1}{q}}&{\text{ if }}x={\frac {p}{q}}{\text{(in lowest terms) is a rational number}}\\0&{\text{ if }}x{\text{ is irrational}}.\end{cases}}}

is continuous at all irrational numbers and discontinuous at all rational numbers. In a similar vein, <a href="/facts/Dirichlet%27s_function/d9o708HR">Dirichlet's function</a>, the <a href="/facts/Indicator_function/QEuh04NM">indicator function</a> for the set of rational numbers,

D
        (
        x
        )
        =
        
          
            {
            
              
                
                  0
                
                
                  
                     if 
                  
                  x
                  
                     is irrational 
                  
                  (
                  ∈
                  
                    R
                  
                  ∖
                  
                    Q
                  
                  )
                
              
              
                
                  1
                
                
                  
                     if 
                  
                  x
                  
                     is rational 
                  
                  (
                  ∈
                  
                    Q
                  
                  )
                
              
            
            
          
        
      
    
    {\displaystyle D(x)={\begin{cases}0&{\text{ if }}x{\text{  is irrational }}(\in \mathbb {R} \setminus \mathbb {Q} )\\1&{\text{ if }}x{\text{ is rational }}(\in \mathbb {Q} )\end{cases}}}

is nowhere continuous.

<h3>Properties</h3>
<h4>A useful lemma</h4>
Let 
 
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(x)}
 
 be a function that is continuous at a point 
 
 
 
 
 x
 
 0
 
 
 ,
 
 
 {\displaystyle x_{0},}
 
 and 
 
 
 
 
 y
 
 0
 
 
 
 
 {\displaystyle y_{0}}
 
 be a value such 
 
 
 
 f
 
 (
 
 x
 
 0
 
 
 )
 
 ≠
 
 y
 
 0
 
 
 .
 
 
 {\displaystyle f\left(x_{0}\right)\neq y_{0}.}
 
 Then 
 
 
 
 f
 (
 x
 )
 ≠
 
 y
 
 0
 
 
 
 
 {\displaystyle f(x)\neq y_{0}}
 
 throughout some neighbourhood of 
 
 
 
 
 x
 
 0
 
 
 .
 
 
 {\displaystyle x_{0}.}
 
<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a>
Proof: By the definition of continuity, take 
 
 
 
 ε
 =
 
 
 
 
 |
 
 
 y
 
 0
 
 
 −
 f
 (
 
 x
 
 0
 
 
 )
 
 |
 
 
 2
 
 
 >
 0
 
 
 {\displaystyle \varepsilon ={\frac {|y_{0}-f(x_{0})|}{2}}>0}
 
 , then there exists 
 
 
 
 δ
 >
 0
 
 
 {\displaystyle \delta >0}
 
 such that

|
 
 f
 (
 x
 )
 −
 f
 (
 
 x
 
 0
 
 
 )
 
 |
 
 <
 
 
 
 |
 
 
 y
 
 0
 
 
 −
 f
 (
 
 x
 
 0
 
 
 )
 
 |
 
 2
 
 
 
 
  whenever 
 
 
 
 |
 
 x
 −
 
 x
 
 0
 
 
 
 |
 
 <
 δ
 
 
 {\displaystyle \left|f(x)-f(x_{0})\right|<{\frac {\left|y_{0}-f(x_{0})\right|}{2}}\quad {\text{ whenever }}\quad |x-x_{0}|<\delta }

Suppose there is a point in the neighbourhood 
 
 
 
 
 |
 
 x
 −
 
 x
 
 0
 
 
 
 |
 
 <
 δ
 
 
 {\displaystyle |x-x_{0}|<\delta }
 
 for which 
 
 
 
 f
 (
 x
 )
 =
 
 y
 
 0
 
 
 ;
 
 
 {\displaystyle f(x)=y_{0};}
 
 then we have the contradiction

|
 
 f
 (
 
 x
 
 0
 
 
 )
 −
 
 y
 
 0
 
 
 
 |
 
 <
 
 
 
 |
 
 f
 (
 
 x
 
 0
 
 
 )
 −
 
 y
 
 0
 
 
 
 |
 
 2
 
 
 .
 
 
 {\displaystyle \left|f(x_{0})-y_{0}\right|<{\frac {\left|f(x_{0})-y_{0}\right|}{2}}.}

<h4>Intermediate value theorem</h4>
The <a href="/facts/Intermediate_value_theorem/SaFdYE7W">intermediate value theorem</a> is an <a href="/facts/Existence_theorem/Au8x1FSt">existence theorem</a>, based on the real number property of <a href="/facts/Real_number/R02gw5Pb">completeness</a>, and states:

If the real-valued function f is continuous on the <a href="/facts/Interval_(mathematics)/e9se3vTe">closed interval</a> 
 
 
 
 [
 a
 ,
 b
 ]
 ,
 
 
 {\displaystyle [a,b],}
 
 and k is some number between 
 
 
 
 f
 (
 a
 )
 
 
 {\displaystyle f(a)}
 
 and 
 
 
 
 f
 (
 b
 )
 ,
 
 
 {\displaystyle f(b),}
 
 then there is some number 
 
 
 
 c
 ∈
 [
 a
 ,
 b
 ]
 ,
 
 
 {\displaystyle c\in [a,b],}
 
 such that 
 
 
 
 f
 (
 c
 )
 =
 k
 .
 
 
 {\displaystyle f(c)=k.}

For example, if a child grows from 1 m to 1.5 m between the ages of two and six years, then, at some time between two and six years of age, the child's height must have been 1.25 m.
As a consequence, if f is continuous on 
 
 
 
 [
 a
 ,
 b
 ]
 
 
 {\displaystyle [a,b]}
 
 and 
 
 
 
 f
 (
 a
 )
 
 
 {\displaystyle f(a)}
 
 and 
 
 
 
 f
 (
 b
 )
 
 
 {\displaystyle f(b)}
 
 differ in <a href="/facts/Sign_(mathematics)/z0xollg1">sign</a>, then, at some point 
 
 
 
 c
 ∈
 [
 a
 ,
 b
 ]
 ,
 
 
 {\displaystyle c\in [a,b],}
 
 
 
 
 
 f
 (
 c
 )
 
 
 {\displaystyle f(c)}
 
 must equal <a href="/facts/0_(number)/4NBRtKHc">zero</a>.

<h4>Extreme value theorem</h4>
The <a href="/facts/Extreme_value_theorem/eSV6P7eh">extreme value theorem</a> states that if a function f is defined on a closed interval 
 
 
 
 [
 a
 ,
 b
 ]
 
 
 {\displaystyle [a,b]}
 
 (or any closed and bounded set) and is continuous there, then the function attains its maximum, i.e. there exists 
 
 
 
 c
 ∈
 [
 a
 ,
 b
 ]
 
 
 {\displaystyle c\in [a,b]}
 
 with 
 
 
 
 f
 (
 c
 )
 ≥
 f
 (
 x
 )
 
 
 {\displaystyle f(c)\geq f(x)}
 
 for all 
 
 
 
 x
 ∈
 [
 a
 ,
 b
 ]
 .
 
 
 {\displaystyle x\in [a,b].}
 
 The same is true of the minimum of f. These statements are not, in general, true if the function is defined on an open interval 
 
 
 
 (
 a
 ,
 b
 )
 
 
 {\displaystyle (a,b)}
 
 (or any set that is not both closed and bounded), as, for example, the continuous function 
 
 
 
 f
 (
 x
 )
 =
 
 
 1
 x
 
 
 ,
 
 
 {\displaystyle f(x)={\frac {1}{x}},}
 
 defined on the open interval (0,1), does not attain a maximum, being unbounded above.

<h4>Relation to differentiability and integrability</h4>
Every <a href="/facts/Differentiable_function/jQqTgLk1">differentiable function</a>

f
        :
        (
        a
        ,
        b
        )
        →
        
          R
        
      
    
    {\displaystyle f:(a,b)\to \mathbb {R} }

is continuous, as can be shown. The <a href="/facts/Theorem/f2Pe1FMD">converse</a> does not hold: for example, the <a href="/facts/Absolute_value/dwJBpEs5">absolute value</a> function

f
 (
 x
 )
 =
 
 |
 
 x
 
 |
 
 =
 
 
 {
 
 
 
 
 
  
 x
 
 
 
  if 
 
 x
 ≥
 0
 
 
 
 
 −
 x
 
 
 
  if 
 
 x
 <
 0
 
 
 
 
 
 
 
 
 {\displaystyle f(x)=|x|={\begin{cases}\;\;\ x&{\text{ if }}x\geq 0\\-x&{\text{ if }}x<0\end{cases}}}

is everywhere continuous. However, it is not differentiable at 
 
 
 
 x
 =
 0
 
 
 {\displaystyle x=0}
 
 (but is so everywhere else). <a href="/facts/Weierstrass_function/m9jU4LE5">Weierstrass's function</a> is also everywhere continuous but nowhere differentiable.
The <a href="/facts/Derivative/7Ito70Dh">derivative</a> f′(x) of a differentiable function f(x) need not be continuous. If f′(x) is continuous, f(x) is said to be continuously differentiable. The set of such functions is denoted 
 
 
 
 
 C
 
 1
 
 
 (
 (
 a
 ,
 b
 )
 )
 .
 
 
 {\displaystyle C^{1}((a,b)).}
 
 More generally, the set of functions

f
        :
        Ω
        →
        
          R
        
      
    
    {\displaystyle f:\Omega \to \mathbb {R} }

(from an open interval (or <a href="/facts/Open_subset/QfTCIqMu">open subset</a> of 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
) 
 
 
 
 Ω
 
 
 {\displaystyle \Omega }
 
 to the reals) such that f is 
 
 
 
 n
 
 
 {\displaystyle n}
 
 times differentiable and such that the 
 
 
 
 n
 
 
 {\displaystyle n}
 
-th derivative of f is continuous is denoted 
 
 
 
 
 C
 
 n
 
 
 (
 Ω
 )
 .
 
 
 {\displaystyle C^{n}(\Omega ).}
 
 See <a href="/facts/Differentiability_class/mffL4ch2">differentiability class</a>. In the field of computer graphics, properties related (but not identical) to 
 
 
 
 
 C
 
 0
 
 
 ,
 
 C
 
 1
 
 
 ,
 
 C
 
 2
 
 
 
 
 {\displaystyle C^{0},C^{1},C^{2}}
 
 are sometimes called 
 
 
 
 
 G
 
 0
 
 
 
 
 {\displaystyle G^{0}}
 
 (continuity of position), 
 
 
 
 
 G
 
 1
 
 
 
 
 {\displaystyle G^{1}}
 
 (continuity of tangency), and 
 
 
 
 
 G
 
 2
 
 
 
 
 {\displaystyle G^{2}}
 
 (continuity of curvature); see <a href="/facts/Smoothness/mffL4ch2">Smoothness of curves and surfaces</a>.
Every continuous function

f
        :
        [
        a
        ,
        b
        ]
        →
        
          R
        
      
    
    {\displaystyle f:[a,b]\to \mathbb {R} }

is <a href="/facts/Integrable_function/V7lyV997">integrable</a> (for example in the sense of the <a href="/facts/Riemann_integral/vWGrPhVh">Riemann integral</a>). The converse does not hold, as the (integrable but discontinuous) <a href="/facts/Sign_function/VADy9zQq">sign function</a> shows.

<h4>Pointwise and uniform limits</h4>

Given a <a href="/facts/Sequence_(mathematics)/gV2S4uqp">sequence</a>

f
          
            1
          
        
        ,
        
          f
          
            2
          
        
        ,
        …
        :
        I
        →
        
          R
        
      
    
    {\displaystyle f_{1},f_{2},\dotsc :I\to \mathbb {R} }

of functions such that the limit

f
        (
        x
        )
        :=
        
          lim
          
            n
            →
            ∞
          
        
        
          f
          
            n
          
        
        (
        x
        )
      
    
    {\displaystyle f(x):=\lim _{n\to \infty }f_{n}(x)}

exists for all 
 
 
 
 x
 ∈
 D
 ,
 
 
 {\displaystyle x\in D,}
 
, the resulting function 
 
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(x)}
 
 is referred to as the <a href="/facts/Pointwise_convergence/TXGpeyIK">pointwise limit</a> of the sequence of functions 
 
 
 
 
 
 (
 
 f
 
 n
 
 
 )
 
 
 n
 ∈
 N
 
 
 .
 
 
 {\displaystyle \left(f_{n}\right)_{n\in N}.}
 
 The pointwise limit function need not be continuous, even if all functions 
 
 
 
 
 f
 
 n
 
 
 
 
 {\displaystyle f_{n}}
 
 are continuous, as the animation at the right shows. However, f is continuous if all functions 
 
 
 
 
 f
 
 n
 
 
 
 
 {\displaystyle f_{n}}
 
 are continuous and the sequence <a href="/facts/Uniform_convergence/ecz9eNWn">converges uniformly</a>, by the <a href="/facts/Uniform_convergence_theorem/ecz9eNWn">uniform convergence theorem</a>. This theorem can be used to show that the <a href="/facts/Exponential_function/ewlYxvR2">exponential functions</a>, <a href="/facts/Logarithm/QeXaV97q">logarithms</a>, <a href="/facts/Square_root/AfrzfBdQ">square root</a> function, and <a href="/facts/Trigonometric_function/czej7ur0">trigonometric functions</a> are continuous.

<h3>Directional Continuity</h3>

Discontinuous functions may be discontinuous in a restricted way, giving rise to the concept of directional continuity (or right and left continuous functions) and <a href="/facts/Semi-continuity/4unJbWdW">semi-continuity</a>. Roughly speaking, a function is right-continuous if no jump occurs when the limit point is approached from the right. Formally, f is said to be right-continuous at the point c if the following holds: For any number 
 
 
 
 ε
 >
 0
 
 
 {\displaystyle \varepsilon >0}
 
 however small, there exists some number 
 
 
 
 δ
 >
 0
 
 
 {\displaystyle \delta >0}
 
 such that for all x in the domain with 
 
 
 
 c
 <
 x
 <
 c
 +
 δ
 ,
 
 
 {\displaystyle c<x<c+\delta ,}
 
 the value of 
 
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(x)}
 
 will satisfy

|
 
 f
 (
 x
 )
 −
 f
 (
 c
 )
 
 |
 
 <
 ε
 .
 
 
 {\displaystyle |f(x)-f(c)|<\varepsilon .}

This is the same condition as continuous functions, except it is required to hold for x strictly larger than c only. Requiring it instead for all x with 
 
 
 
 c
 −
 δ
 <
 x
 <
 c
 
 
 {\displaystyle c-\delta <x<c}
 
 yields the notion of left-continuous functions. A function is continuous if and only if it is both right-continuous and left-continuous.

<h3>Semicontinuity</h3>
Main article: <a href="/facts/Semicontinuity/4unJbWdW">Semicontinuity</a>
A function f is <a href="/facts/Semi-continuity/4unJbWdW">lower semi-continuous</a> if, roughly, any jumps that might occur only go down, but not up. That is, for any 
 
 
 
 ε
 >
 0
 ,
 
 
 {\displaystyle \varepsilon >0,}
 
 there exists some number 
 
 
 
 δ
 >
 0
 
 
 {\displaystyle \delta >0}
 
 such that for all x in the domain with 
 
 
 
 
 |
 
 x
 −
 c
 
 |
 
 <
 δ
 ,
 
 
 {\displaystyle |x-c|<\delta ,}
 
 the value of 
 
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(x)}
 
 satisfies

f
        (
        x
        )
        ≥
        f
        (
        c
        )
        −
        ϵ
        .
      
    
    {\displaystyle f(x)\geq f(c)-\epsilon .}

The reverse condition is <a href="/facts/Semi-continuity/4unJbWdW">upper semi-continuity</a>.

<h2 id="continuous-functions-between-metric-spaces">Continuous functions between metric spaces</h2>
The concept of continuous real-valued functions can be generalized to functions between <a href="/facts/Metric_space/gnBBRXtc">metric spaces</a>. A metric space is a set 
 
 
 
 X
 
 
 {\displaystyle X}
 
 equipped with a function (called <a href="/facts/Metric_(mathematics)/RkUptSo8">metric</a>) 
 
 
 
 
 d
 
 X
 
 
 ,
 
 
 {\displaystyle d_{X},}
 
 that can be thought of as a measurement of the distance of any two elements in X. Formally, the metric is a function

d
          
            X
          
        
        :
        X
        ×
        X
        →
        
          R
        
      
    
    {\displaystyle d_{X}:X\times X\to \mathbb {R} }

that satisfies a number of requirements, notably the <a href="/facts/Triangle_inequality/KubLjkwr">triangle inequality</a>. Given two metric spaces 
 
 
 
 
 (
 
 X
 ,
 
 d
 
 X
 
 
 
 )
 
 
 
 {\displaystyle \left(X,d_{X}\right)}
 
 and 
 
 
 
 
 (
 
 Y
 ,
 
 d
 
 Y
 
 
 
 )
 
 
 
 {\displaystyle \left(Y,d_{Y}\right)}
 
 and a function

f
        :
        X
        →
        Y
      
    
    {\displaystyle f:X\to Y}

then 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is continuous at the point 
 
 
 
 c
 ∈
 X
 
 
 {\displaystyle c\in X}
 
 (with respect to the given metrics) if for any positive real number 
 
 
 
 ε
 >
 0
 ,
 
 
 {\displaystyle \varepsilon >0,}
 
 there exists a positive real number 
 
 
 
 δ
 >
 0
 
 
 {\displaystyle \delta >0}
 
 such that all 
 
 
 
 x
 ∈
 X
 
 
 {\displaystyle x\in X}
 
 satisfying 
 
 
 
 
 d
 
 X
 
 
 (
 x
 ,
 c
 )
 <
 δ
 
 
 {\displaystyle d_{X}(x,c)<\delta }
 
 will also satisfy 
 
 
 
 
 d
 
 Y
 
 
 (
 f
 (
 x
 )
 ,
 f
 (
 c
 )
 )
 <
 ε
 .
 
 
 {\displaystyle d_{Y}(f(x),f(c))<\varepsilon .}
 
 As in the case of real functions above, this is equivalent to the condition that for every sequence 
 
 
 
 
 (
 
 x
 
 n
 
 
 )
 
 
 
 {\displaystyle \left(x_{n}\right)}
 
 in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 with limit 
 
 
 
 lim
 
 x
 
 n
 
 
 =
 c
 ,
 
 
 {\displaystyle \lim x_{n}=c,}
 
 we have 
 
 
 
 lim
 f
 
 (
 
 x
 
 n
 
 
 )
 
 =
 f
 (
 c
 )
 .
 
 
 {\displaystyle \lim f\left(x_{n}\right)=f(c).}
 
 The latter condition can be weakened as follows: 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is continuous at the point 
 
 
 
 c
 
 
 {\displaystyle c}
 
 if and only if for every convergent sequence 
 
 
 
 
 (
 
 x
 
 n
 
 
 )
 
 
 
 {\displaystyle \left(x_{n}\right)}
 
 in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 with limit 
 
 
 
 c
 
 
 {\displaystyle c}
 
, the sequence 
 
 
 
 
 (
 
 f
 
 (
 
 x
 
 n
 
 
 )
 
 
 )
 
 
 
 {\displaystyle \left(f\left(x_{n}\right)\right)}
 
 is a <a href="/facts/Cauchy_sequence/7Q5c8TuF">Cauchy sequence</a>, and 
 
 
 
 c
 
 
 {\displaystyle c}
 
 is in the domain of 
 
 
 
 f
 
 
 {\displaystyle f}
 
.
The set of points at which a function between metric spaces is continuous is a <a href="/facts/G%CE%B4_set/Erg1BCFk">
 
 
 
 
 G
 
 δ
 
 
 
 
 {\displaystyle G_{\delta }}
 
 set</a> – this follows from the 
 
 
 
 ε
 −
 δ
 
 
 {\displaystyle \varepsilon -\delta }
 
 definition of continuity.
This notion of continuity is applied, for example, in <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a>. A key statement in this area says that a <a href="/facts/Linear_operator/Q1PBKNVV">linear operator</a>

T
        :
        V
        →
        W
      
    
    {\displaystyle T:V\to W}

between <a href="/facts/Normed_vector_space/rmDljfyE">normed vector spaces</a> 
 
 
 
 V
 
 
 {\displaystyle V}
 
 and 
 
 
 
 W
 
 
 {\displaystyle W}
 
 (which are <a href="/facts/Vector_space/M1pxMLx2">vector spaces</a> equipped with a compatible <a href="/facts/Norm_(mathematics)/xIbR4uE1">norm</a>, denoted 
 
 
 
 ‖
 x
 ‖
 
 
 {\displaystyle \|x\|}
 
) is continuous if and only if it is <a href="/facts/Bounded_linear_operator/LYmDTIqR">bounded</a>, that is, there is a constant 
 
 
 
 K
 
 
 {\displaystyle K}
 
 such that

‖
        T
        (
        x
        )
        ‖
        ≤
        K
        ‖
        x
        ‖
      
    
    {\displaystyle \|T(x)\|\leq K\|x\|}

for all 
  
    
      
        x
        ∈
        V
        .
      
    
    {\displaystyle x\in V.}

<h3>Uniform, Hölder and Lipschitz continuity</h3>

The concept of continuity for functions between metric spaces can be strengthened in various ways by limiting the way 
 
 
 
 δ
 
 
 {\displaystyle \delta }
 
 depends on 
 
 
 
 ε
 
 
 {\displaystyle \varepsilon }
 
 and c in the definition above. Intuitively, a function f as above is <a href="/facts/Uniformly_continuous/hjvw7tII">uniformly continuous</a> if the 
 
 
 
 δ
 
 
 {\displaystyle \delta }
 
 does
not depend on the point c. More precisely, it is required that for every <a href="/facts/Real_number/R02gw5Pb">real number</a> 
 
 
 
 ε
 >
 0
 
 
 {\displaystyle \varepsilon >0}
 
 there exists 
 
 
 
 δ
 >
 0
 
 
 {\displaystyle \delta >0}
 
 such that for every 
 
 
 
 c
 ,
 b
 ∈
 X
 
 
 {\displaystyle c,b\in X}
 
 with 
 
 
 
 
 d
 
 X
 
 
 (
 b
 ,
 c
 )
 <
 δ
 ,
 
 
 {\displaystyle d_{X}(b,c)<\delta ,}
 
 we have that 
 
 
 
 
 d
 
 Y
 
 
 (
 f
 (
 b
 )
 ,
 f
 (
 c
 )
 )
 <
 ε
 .
 
 
 {\displaystyle d_{Y}(f(b),f(c))<\varepsilon .}
 
 Thus, any uniformly continuous function is continuous. The converse does not generally hold but holds when the domain space X is <a href="/facts/Compact_topological_space/d0cgXJH7">compact</a>. Uniformly continuous maps can be defined in the more general situation of <a href="/facts/Uniform_space/GKumHlHo">uniform spaces</a>.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a>
A function is <a href="/facts/H%C3%B6lder_continuity/CI0yp6kD">Hölder continuous</a> with exponent α (a real number) if there is a constant K such that for all 
 
 
 
 b
 ,
 c
 ∈
 X
 ,
 
 
 {\displaystyle b,c\in X,}
 
 the inequality

d
          
            Y
          
        
        (
        f
        (
        b
        )
        ,
        f
        (
        c
        )
        )
        ≤
        K
        ⋅
        (
        
          d
          
            X
          
        
        (
        b
        ,
        c
        )
        
          )
          
            α
          
        
      
    
    {\displaystyle d_{Y}(f(b),f(c))\leq K\cdot (d_{X}(b,c))^{\alpha }}

holds. Any Hölder continuous function is uniformly continuous. The particular case 
 
 
 
 α
 =
 1
 
 
 {\displaystyle \alpha =1}
 
 is referred to as <a href="/facts/Lipschitz_continuity/Hw20EPEU">Lipschitz continuity</a>. That is, a function is Lipschitz continuous if there is a constant K such that the inequality

d
          
            Y
          
        
        (
        f
        (
        b
        )
        ,
        f
        (
        c
        )
        )
        ≤
        K
        ⋅
        
          d
          
            X
          
        
        (
        b
        ,
        c
        )
      
    
    {\displaystyle d_{Y}(f(b),f(c))\leq K\cdot d_{X}(b,c)}

holds for any 
 
 
 
 b
 ,
 c
 ∈
 X
 .
 
 
 {\displaystyle b,c\in X.}
 
<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a> The Lipschitz condition occurs, for example, in the <a href="/facts/Picard%E2%80%93Lindel%C3%B6f_theorem/yKHSWoMJ">Picard–Lindelöf theorem</a> concerning the solutions of <a href="/facts/Ordinary_differential_equation/ygKGQ8kD">ordinary differential equations</a>.

<h2 id="continuous-functions-between-topological-spaces">Continuous functions between topological spaces</h2>
Another, more abstract, notion of continuity is the continuity of functions between <a href="/facts/Topological_space/v2ut7CbK">topological spaces</a> in which there generally is no formal notion of distance, as there is in the case of <a href="/facts/Metric_space/gnBBRXtc">metric spaces</a>. A topological space is a set X together with a topology on X, which is a set of <a href="/facts/Subset/SJGERmbA">subsets</a> of X satisfying a few requirements with respect to their unions and intersections that generalize the properties of the <a href="/facts/Open_ball/f9vpMVMp">open balls</a> in metric spaces while still allowing one to talk about the <a href="/facts/Neighborhood_(mathematics)/XHKgrpkp"> neighborhoods</a> of a given point. The elements of a topology are called <a href="/facts/Open_subset/QfTCIqMu">open subsets</a> of X (with respect to the topology).
A function

f
        :
        X
        →
        Y
      
    
    {\displaystyle f:X\to Y}

between two topological spaces X and Y is continuous if for every open set 
 
 
 
 V
 ⊆
 Y
 ,
 
 
 {\displaystyle V\subseteq Y,}
 
 the <a href="/facts/Image_(mathematics)/KANefMAl">inverse image</a>

f
          
            −
            1
          
        
        (
        V
        )
        =
        {
        x
        ∈
        X
        
        
          |
        
        
        f
        (
        x
        )
        ∈
        V
        }
      
    
    {\displaystyle f^{-1}(V)=\{x\in X\;|\;f(x)\in V\}}

is an open subset of X. That is, f is a function between the sets X and Y (not on the elements of the topology 
 
 
 
 
 T
 
 X
 
 
 
 
 {\displaystyle T_{X}}
 
), but the continuity of f depends on the topologies used on X and Y.
This is equivalent to the condition that the <a href="/facts/Image_(mathematics)/KANefMAl">preimages</a> of the <a href="/facts/Closed_set/upYnRTUj">closed sets</a> (which are the complements of the open subsets) in Y are closed in X.
An extreme example: if a set X is given the <a href="/facts/Discrete_topology/hpvXA23u">discrete topology</a> (in which every subset is open), all functions

f
        :
        X
        →
        T
      
    
    {\displaystyle f:X\to T}

to any topological space T are continuous. On the other hand, if X is equipped with the <a href="/facts/Indiscrete_topology/uA7RlGth">indiscrete topology</a> (in which the only open subsets are the empty set and X) and the space T set is at least <a href="/facts/T0_space/dk7qNoRn">T0</a>, then the only continuous functions are the constant functions. Conversely, any function whose codomain is indiscrete is continuous.

<h3>Continuity at a point</h3>

The translation in the language of neighborhoods of the <a href="/facts/(%CE%B5,_%CE%B4)-definition_of_limit/3Yd7BdvM">
 
 
 
 (
 ε
 ,
 δ
 )
 
 
 {\displaystyle (\varepsilon ,\delta )}
 
-definition of continuity</a> leads to the following definition of the continuity at a point:

A function 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\to Y}
 
 is continuous at a point 
 
 
 
 x
 ∈
 X
 
 
 {\displaystyle x\in X}
 
 if and only if for any neighborhood V of 
 
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(x)}
 
 in Y, there is a neighborhood U of 
 
 
 
 x
 
 
 {\displaystyle x}
 
 such that 
 
 
 
 f
 (
 U
 )
 ⊆
 V
 .
 
 
 {\displaystyle f(U)\subseteq V.}

This definition is equivalent to the same statement with neighborhoods restricted to open neighborhoods and can be restated in several ways by using <a href="/facts/Preimage/KANefMAl">preimages</a> rather than images.
Also, as every set that contains a neighborhood is also a neighborhood, and 
 
 
 
 
 f
 
 −
 1
 
 
 (
 V
 )
 
 
 {\displaystyle f^{-1}(V)}
 
 is the largest subset U of X such that 
 
 
 
 f
 (
 U
 )
 ⊆
 V
 ,
 
 
 {\displaystyle f(U)\subseteq V,}
 
 this definition may be simplified into:

A function 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\to Y}
 
 is continuous at a point 
 
 
 
 x
 ∈
 X
 
 
 {\displaystyle x\in X}
 
 if and only if 
 
 
 
 
 f
 
 −
 1
 
 
 (
 V
 )
 
 
 {\displaystyle f^{-1}(V)}
 
 is a neighborhood of 
 
 
 
 x
 
 
 {\displaystyle x}
 
 for every neighborhood V of 
 
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(x)}
 
 in Y.
As an open set is a set that is a neighborhood of all its points, a function 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\to Y}
 
 is continuous at every point of X if and only if it is a continuous function.
If X and Y are metric spaces, it is equivalent to consider the <a href="/facts/Neighborhood_system/fMRM8Xht">neighborhood system</a> of <a href="/facts/Open_ball/f9vpMVMp">open balls</a> centered at x and f(x) instead of all neighborhoods. This gives back the above 
 
 
 
 ε
 −
 δ
 
 
 {\displaystyle \varepsilon -\delta }
 
 definition of continuity in the context of metric spaces. In general topological spaces, there is no notion of nearness or distance. If, however, the target space is a <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff space</a>, it is still true that f is continuous at a if and only if the limit of f as x approaches a is f(a). At an isolated point, every function is continuous.
Given 
 
 
 
 x
 ∈
 X
 ,
 
 
 {\displaystyle x\in X,}
 
 a map 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\to Y}
 
 is continuous at 
 
 
 
 x
 
 
 {\displaystyle x}
 
 if and only if whenever 
 
 
 
 
 
 B
 
 
 
 
 {\displaystyle {\mathcal {B}}}
 
 is a filter on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 that <a href="/facts/Convergent_filter/5vQbHcpE">converges</a> to 
 
 
 
 x
 
 
 {\displaystyle x}
 
 in 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 which is expressed by writing 
 
 
 
 
 
 B
 
 
 →
 x
 ,
 
 
 {\displaystyle {\mathcal {B}}\to x,}
 
 then necessarily 
 
 
 
 f
 (
 
 
 B
 
 
 )
 →
 f
 (
 x
 )
 
 
 {\displaystyle f({\mathcal {B}})\to f(x)}
 
 in 
 
 
 
 Y
 .
 
 
 {\displaystyle Y.}
 
 
If 
 
 
 
 
 
 N
 
 
 (
 x
 )
 
 
 {\displaystyle {\mathcal {N}}(x)}
 
 denotes the <a href="/facts/Neighborhood_filter/fMRM8Xht">neighborhood filter</a> at 
 
 
 
 x
 
 
 {\displaystyle x}
 
 then 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\to Y}
 
 is continuous at 
 
 
 
 x
 
 
 {\displaystyle x}
 
 if and only if 
 
 
 
 f
 (
 
 
 N
 
 
 (
 x
 )
 )
 →
 f
 (
 x
 )
 
 
 {\displaystyle f({\mathcal {N}}(x))\to f(x)}
 
 in 
 
 
 
 Y
 .
 
 
 {\displaystyle Y.}
 
<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a> Moreover, this happens if and only if the <a href="/facts/Prefilter/7f3zk55u">prefilter</a> 
 
 
 
 f
 (
 
 
 N
 
 
 (
 x
 )
 )
 
 
 {\displaystyle f({\mathcal {N}}(x))}
 
 is a <a href="/facts/Filter_base/7f3zk55u">filter base</a> for the neighborhood filter of 
 
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(x)}
 
 in 
 
 
 
 Y
 .
 
 
 {\displaystyle Y.}
 
<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a>

<h3>Alternative definitions</h3>
Several <a href="/facts/Characterizations_of_the_category_of_topological_spaces/moIyGjVd">equivalent definitions for a topological structure</a> exist; thus, several equivalent ways exist to define a continuous function.

<h4>Sequences and nets </h4>
In several contexts, the topology of a space is conveniently specified in terms of <a href="/facts/Limit_points/95MykoGU">limit points</a>. This is often accomplished by specifying when a point is the <a href="/facts/Limit_of_a_sequence/Ktge2RwL">limit of a sequence</a>. Still, for some spaces that are too large in some sense, one specifies also when a point is the limit of more general sets of points <a href="/facts/Indexed_family/aFqfzJN0">indexed</a> by a <a href="/facts/Directed_set/GmWcqe3L">directed set</a>, known as <a href="/facts/Net_(mathematics)/msmsvMTT">nets</a>. A function is (Heine-)continuous only if it takes limits of sequences to limits of sequences. In the former case, preservation of limits is also sufficient; in the latter, a function may preserve all limits of sequences yet still fail to be continuous, and preservation of nets is a necessary and sufficient condition.
In detail, a function 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\to Y}
 
 is <a href="/facts/Sequential_continuity/xAjFpdfR">sequentially continuous</a> if whenever a sequence 
 
 
 
 
 (
 
 x
 
 n
 
 
 )
 
 
 
 {\displaystyle \left(x_{n}\right)}
 
 in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 converges to a limit 
 
 
 
 x
 ,
 
 
 {\displaystyle x,}
 
 the sequence 
 
 
 
 
 (
 
 f
 
 (
 
 x
 
 n
 
 
 )
 
 
 )
 
 
 
 {\displaystyle \left(f\left(x_{n}\right)\right)}
 
 converges to 
 
 
 
 f
 (
 x
 )
 .
 
 
 {\displaystyle f(x).}
 
 Thus, sequentially continuous functions "preserve sequential limits." Every continuous function is sequentially continuous. If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a <a href="/facts/First-countable_space/HDuTYa6A">first-countable space</a> and <a href="/facts/Axiom_of_countable_choice/goSPxEdj">countable choice</a> holds, then the converse also holds: any function preserving sequential limits is continuous. In particular, if 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a metric space, sequential continuity and continuity are equivalent. For non-first-countable spaces, sequential continuity might be strictly weaker than continuity. (The spaces for which the two properties are equivalent are called <a href="/facts/Sequential_space/xAjFpdfR">sequential spaces</a>.) This motivates the consideration of nets instead of sequences in general topological spaces. Continuous functions preserve the limits of nets, and this property characterizes continuous functions.
For instance, consider the case of real-valued functions of one real variable:<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a>

Theorem—A function 
 
 
 
 f
 :
 A
 ⊆
 
 R
 
 →
 
 R
 
 
 
 {\displaystyle f:A\subseteq \mathbb {R} \to \mathbb {R} }
 
 is continuous at 
 
 
 
 
 x
 
 0
 
 
 
 
 {\displaystyle x_{0}}
 
 if and only if it is <a href="/facts/Sequentially_continuous/xAjFpdfR">sequentially continuous</a> at that point.

<table><tbody><tr><th>Proof</th></tr><tr><td>Proof. Assume that f : A ⊆ R → R {\displaystyle f:A\subseteq \mathbb {R} \to \mathbb {R} } is continuous at x 0 {\displaystyle x_{0}} (in the sense of <a href="/facts/(%CE%B5,_%CE%B4)-definition_of_limit/3Yd7BdvM"> ϵ − δ {\displaystyle \epsilon -\delta } continuity</a>). Let ( x n ) n ≥ 1 {\displaystyle \left(x_{n}\right)_{n\geq 1}} be a sequence converging at x 0 {\displaystyle x_{0}} (such a sequence always exists, for example, x n = x ,  for all  n {\displaystyle x_{n}=x,{\text{ for all }}n} ); since f {\displaystyle f} is continuous at x 0 {\displaystyle x_{0}} ∀ ϵ > 0 ∃ δ ϵ > 0 : 0 < | x − x 0 | < δ ϵ ⟹ | f ( x ) − f ( x 0 ) | < ϵ . ( ∗ ) {\displaystyle \forall \epsilon >0\,\exists \delta _{\epsilon }>0:0<|x-x_{0}|<\delta _{\epsilon }\implies |f(x)-f(x_{0})|<\epsilon .\quad (*)} For any such δ ϵ {\displaystyle \delta _{\epsilon }} we can find a natural number ν ϵ > 0 {\displaystyle \nu _{\epsilon }>0} such that for all n > ν ϵ , {\displaystyle n>\nu _{\epsilon },} | x n − x 0 | < δ ϵ , {\displaystyle |x_{n}-x_{0}|<\delta _{\epsilon },} since ( x n ) {\displaystyle \left(x_{n}\right)} converges at x 0 {\displaystyle x_{0}} ; combining this with ( ∗ ) {\displaystyle (*)} we obtain ∀ ϵ > 0 ∃ ν ϵ > 0 : ∀ n > ν ϵ | f ( x n ) − f ( x 0 ) | < ϵ . {\displaystyle \forall \epsilon >0\,\exists \nu _{\epsilon }>0:\forall n>\nu _{\epsilon }\quad |f(x_{n})-f(x_{0})|<\epsilon .} Assume on the contrary that f {\displaystyle f} is sequentially continuous and proceed by contradiction: suppose f {\displaystyle f} is not continuous at x 0 {\displaystyle x_{0}} ∃ ϵ > 0 : ∀ δ ϵ > 0 , ∃ x δ ϵ : 0 < | x δ ϵ − x 0 | < δ ϵ ⟹ | f ( x δ ϵ ) − f ( x 0 ) | > ϵ {\displaystyle \exists \epsilon >0:\forall \delta _{\epsilon }>0,\,\exists x_{\delta _{\epsilon }}:0<|x_{\delta _{\epsilon }}-x_{0}|<\delta _{\epsilon }\implies |f(x_{\delta _{\epsilon }})-f(x_{0})|>\epsilon } then we can take δ ϵ = 1 / n , ∀ n > 0 {\displaystyle \delta _{\epsilon }=1/n,\,\forall n>0} and call the corresponding point x δ ϵ =: x n {\displaystyle x_{\delta _{\epsilon }}=:x_{n}} : in this way we have defined a sequence ( x n ) n ≥ 1 {\displaystyle (x_{n})_{n\geq 1}} such that ∀ n > 0 | x n − x 0 | < 1 n , | f ( x n ) − f ( x 0 ) | > ϵ {\displaystyle \forall n>0\quad |x_{n}-x_{0}|<{\frac {1}{n}},\quad |f(x_{n})-f(x_{0})|>\epsilon } by construction x n → x 0 {\displaystyle x_{n}\to x_{0}} but f ( x n ) ↛ f ( x 0 ) {\displaystyle f(x_{n})\not \to f(x_{0})} , which contradicts the hypothesis of sequential continuity. ◼ {\displaystyle \blacksquare } </td></tr></tbody></table>
<h4>Closure operator and interior operator definitions</h4>
In terms of the <a href="/facts/Interior_(topology)/5sxGnQfY">interior</a> operator, a function 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\to Y}
 
 between topological spaces is continuous if and only if for every subset 
 
 
 
 B
 ⊆
 Y
 ,
 
 
 {\displaystyle B\subseteq Y,}

f
          
            −
            1
          
        
        
          (
          
            
              int
              
                Y
              
            
            ⁡
            B
          
          )
        
         
        ⊆
         
        
          int
          
            X
          
        
        ⁡
        
          (
          
            
              f
              
                −
                1
              
            
            (
            B
            )
          
          )
        
        .
      
    
    {\displaystyle f^{-1}\left(\operatorname {int} _{Y}B\right)~\subseteq ~\operatorname {int} _{X}\left(f^{-1}(B)\right).}

In terms of the <a href="/facts/Closure_(topology)/9x2wlgUW">closure</a> operator, 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\to Y}
 
 is continuous if and only if for every subset 
 
 
 
 A
 ⊆
 X
 ,
 
 
 {\displaystyle A\subseteq X,}

f
        
          (
          
            
              cl
              
                X
              
            
            ⁡
            A
          
          )
        
         
        ⊆
         
        
          cl
          
            Y
          
        
        ⁡
        (
        f
        (
        A
        )
        )
        .
      
    
    {\displaystyle f\left(\operatorname {cl} _{X}A\right)~\subseteq ~\operatorname {cl} _{Y}(f(A)).}

That is to say, given any element 
 
 
 
 x
 ∈
 X
 
 
 {\displaystyle x\in X}
 
 that belongs to the closure of a subset 
 
 
 
 A
 ⊆
 X
 ,
 
 
 {\displaystyle A\subseteq X,}
 
 
 
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(x)}
 
 necessarily belongs to the closure of 
 
 
 
 f
 (
 A
 )
 
 
 {\displaystyle f(A)}
 
 in 
 
 
 
 Y
 .
 
 
 {\displaystyle Y.}
 
 If we declare that a point 
 
 
 
 x
 
 
 {\displaystyle x}
 
 is close to a subset 
 
 
 
 A
 ⊆
 X
 
 
 {\displaystyle A\subseteq X}
 
 if 
 
 
 
 x
 ∈
 
 cl
 
 X
 
 
 ⁡
 A
 ,
 
 
 {\displaystyle x\in \operatorname {cl} _{X}A,}
 
 then this terminology allows for a <a href="/facts/Plain_English/tJaBM1z5">plain English</a> description of continuity: 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is continuous if and only if for every subset 
 
 
 
 A
 ⊆
 X
 ,
 
 
 {\displaystyle A\subseteq X,}
 
 
 
 
 
 f
 
 
 {\displaystyle f}
 
 maps points that are close to 
 
 
 
 A
 
 
 {\displaystyle A}
 
 to points that are close to 
 
 
 
 f
 (
 A
 )
 .
 
 
 {\displaystyle f(A).}
 
 Similarly, 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is continuous at a fixed given point 
 
 
 
 x
 ∈
 X
 
 
 {\displaystyle x\in X}
 
 if and only if whenever 
 
 
 
 x
 
 
 {\displaystyle x}
 
 is close to a subset 
 
 
 
 A
 ⊆
 X
 ,
 
 
 {\displaystyle A\subseteq X,}
 
 then 
 
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(x)}
 
 is close to 
 
 
 
 f
 (
 A
 )
 .
 
 
 {\displaystyle f(A).}

Instead of specifying topological spaces by their <a href="/facts/Open_set/QfTCIqMu">open subsets</a>, any topology on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 can <a href="/facts/Equivalence_of_categories/k6uzqAcy">alternatively be determined</a> by a <a href="/facts/Kuratowski_closure_operator/ovjzpSms">closure operator</a> or by an <a href="/facts/Interior_operator/Ysc3N0co">interior operator</a>. 
Specifically, the map that sends a subset 
 
 
 
 A
 
 
 {\displaystyle A}
 
 of a topological space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 to its <a href="/facts/Closure_(topology)/9x2wlgUW">topological closure</a> 
 
 
 
 
 cl
 
 X
 
 
 ⁡
 A
 
 
 {\displaystyle \operatorname {cl} _{X}A}
 
 satisfies the <a href="/facts/Kuratowski_closure_axioms/ovjzpSms">Kuratowski closure axioms</a>. Conversely, for any <a href="/facts/Kuratowski_closure_operator/ovjzpSms">closure operator</a> 
 
 
 
 A
 ↦
 cl
 ⁡
 A
 
 
 {\displaystyle A\mapsto \operatorname {cl} A}
 
 there exists a unique topology 
 
 
 
 τ
 
 
 {\displaystyle \tau }
 
 on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 (specifically, 
 
 
 
 τ
 :=
 {
 X
 ∖
 cl
 ⁡
 A
 :
 A
 ⊆
 X
 }
 
 
 {\displaystyle \tau :=\{X\setminus \operatorname {cl} A:A\subseteq X\}}
 
) such that for every subset 
 
 
 
 A
 ⊆
 X
 ,
 
 
 {\displaystyle A\subseteq X,}
 
 
 
 
 
 cl
 ⁡
 A
 
 
 {\displaystyle \operatorname {cl} A}
 
 is equal to the topological closure 
 
 
 
 
 cl
 
 (
 X
 ,
 τ
 )
 
 
 ⁡
 A
 
 
 {\displaystyle \operatorname {cl} _{(X,\tau )}A}
 
 of 
 
 
 
 A
 
 
 {\displaystyle A}
 
 in 
 
 
 
 (
 X
 ,
 τ
 )
 .
 
 
 {\displaystyle (X,\tau ).}
 
 If the sets 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are each associated with closure operators (both denoted by 
 
 
 
 cl
 
 
 {\displaystyle \operatorname {cl} }
 
) then a map 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\to Y}
 
 is continuous if and only if 
 
 
 
 f
 (
 cl
 ⁡
 A
 )
 ⊆
 cl
 ⁡
 (
 f
 (
 A
 )
 )
 
 
 {\displaystyle f(\operatorname {cl} A)\subseteq \operatorname {cl} (f(A))}
 
 for every subset 
 
 
 
 A
 ⊆
 X
 .
 
 
 {\displaystyle A\subseteq X.}

Similarly, the map that sends a subset 
 
 
 
 A
 
 
 {\displaystyle A}
 
 of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 to its <a href="/facts/Interior_(topology)/5sxGnQfY">topological interior</a> 
 
 
 
 
 int
 
 X
 
 
 ⁡
 A
 
 
 {\displaystyle \operatorname {int} _{X}A}
 
 defines an <a href="/facts/Interior_operator/Ysc3N0co">interior operator</a>. Conversely, any interior operator 
 
 
 
 A
 ↦
 int
 ⁡
 A
 
 
 {\displaystyle A\mapsto \operatorname {int} A}
 
 induces a unique topology 
 
 
 
 τ
 
 
 {\displaystyle \tau }
 
 on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 (specifically, 
 
 
 
 τ
 :=
 {
 int
 ⁡
 A
 :
 A
 ⊆
 X
 }
 
 
 {\displaystyle \tau :=\{\operatorname {int} A:A\subseteq X\}}
 
) such that for every 
 
 
 
 A
 ⊆
 X
 ,
 
 
 {\displaystyle A\subseteq X,}
 
 
 
 
 
 int
 ⁡
 A
 
 
 {\displaystyle \operatorname {int} A}
 
 is equal to the topological interior 
 
 
 
 
 int
 
 (
 X
 ,
 τ
 )
 
 
 ⁡
 A
 
 
 {\displaystyle \operatorname {int} _{(X,\tau )}A}
 
 of 
 
 
 
 A
 
 
 {\displaystyle A}
 
 in 
 
 
 
 (
 X
 ,
 τ
 )
 .
 
 
 {\displaystyle (X,\tau ).}
 
 If the sets 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are each associated with interior operators (both denoted by 
 
 
 
 int
 
 
 {\displaystyle \operatorname {int} }
 
) then a map 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\to Y}
 
 is continuous if and only if 
 
 
 
 
 f
 
 −
 1
 
 
 (
 int
 ⁡
 B
 )
 ⊆
 int
 ⁡
 
 (
 
 
 f
 
 −
 1
 
 
 (
 B
 )
 
 )
 
 
 
 {\displaystyle f^{-1}(\operatorname {int} B)\subseteq \operatorname {int} \left(f^{-1}(B)\right)}
 
 for every subset 
 
 
 
 B
 ⊆
 Y
 .
 
 
 {\displaystyle B\subseteq Y.}
 
<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a>

<h4>Filters and prefilters</h4>
Main article: <a href="/facts/Filters_in_topology/5vQbHcpE">Filters in topology</a>
Continuity can also be characterized in terms of <a href="/facts/Filter_(set_theory)/7f3zk55u">filters</a>. A function 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\to Y}
 
 is continuous if and only if whenever a filter 
 
 
 
 
 
 B
 
 
 
 
 {\displaystyle {\mathcal {B}}}
 
 on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 <a href="/facts/Convergent_filter/5vQbHcpE">converges</a> in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 to a point 
 
 
 
 x
 ∈
 X
 ,
 
 
 {\displaystyle x\in X,}
 
 then the <a href="/facts/Prefilter/7f3zk55u">prefilter</a> 
 
 
 
 f
 (
 
 
 B
 
 
 )
 
 
 {\displaystyle f({\mathcal {B}})}
 
 converges in 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 to 
 
 
 
 f
 (
 x
 )
 .
 
 
 {\displaystyle f(x).}
 
 This characterization remains true if the word "filter" is replaced by "prefilter."<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a>

<h3>Properties</h3>
If 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\to Y}
 
 and 
 
 
 
 g
 :
 Y
 →
 Z
 
 
 {\displaystyle g:Y\to Z}
 
 are continuous, then so is the composition 
 
 
 
 g
 ∘
 f
 :
 X
 →
 Z
 .
 
 
 {\displaystyle g\circ f:X\to Z.}
 
 If 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\to Y}
 
 is continuous and

<ul><li>X is <a href="/facts/Compact_space/d0cgXJH7">compact</a>, then f(X) is compact.</li>
<li>X is <a href="/facts/Connected_space/5CUTxfoA">connected</a>, then f(X) is connected.</li>
<li>X is <a href="/facts/Path-connected/5CUTxfoA">path-connected</a>, then f(X) is path-connected.</li>
<li>X is <a href="/facts/Lindel%C3%B6f_space/HC4t6bvz">Lindelöf</a>, then f(X) is Lindelöf.</li>
<li>X is <a href="/facts/Separable_space/2bmU73bk">separable</a>, then f(X) is separable.</li></ul>
The possible topologies on a fixed set X are <a href="/facts/Partial_ordering/qb4a3FAX">partially ordered</a>: a topology 
 
 
 
 
 τ
 
 1
 
 
 
 
 {\displaystyle \tau _{1}}
 
 is said to be <a href="/facts/Comparison_of_topologies/5Q9F2fjf">coarser</a> than another topology 
 
 
 
 
 τ
 
 2
 
 
 
 
 {\displaystyle \tau _{2}}
 
 (notation: 
 
 
 
 
 τ
 
 1
 
 
 ⊆
 
 τ
 
 2
 
 
 
 
 {\displaystyle \tau _{1}\subseteq \tau _{2}}
 
) if every open subset with respect to 
 
 
 
 
 τ
 
 1
 
 
 
 
 {\displaystyle \tau _{1}}
 
 is also open with respect to 
 
 
 
 
 τ
 
 2
 
 
 .
 
 
 {\displaystyle \tau _{2}.}
 
 Then, the <a href="/facts/Identity_function/9AtsP0LC">identity map</a>

id
          
            X
          
        
        :
        
          (
          
            X
            ,
            
              τ
              
                2
              
            
          
          )
        
        →
        
          (
          
            X
            ,
            
              τ
              
                1
              
            
          
          )
        
      
    
    {\displaystyle \operatorname {id} _{X}:\left(X,\tau _{2}\right)\to \left(X,\tau _{1}\right)}

is continuous if and only if 
 
 
 
 
 τ
 
 1
 
 
 ⊆
 
 τ
 
 2
 
 
 
 
 {\displaystyle \tau _{1}\subseteq \tau _{2}}
 
 (see also <a href="/facts/Comparison_of_topologies/5Q9F2fjf">comparison of topologies</a>). More generally, a continuous function

(
          
            X
            ,
            
              τ
              
                X
              
            
          
          )
        
        →
        
          (
          
            Y
            ,
            
              τ
              
                Y
              
            
          
          )
        
      
    
    {\displaystyle \left(X,\tau _{X}\right)\to \left(Y,\tau _{Y}\right)}

stays continuous if the topology 
 
 
 
 
 τ
 
 Y
 
 
 
 
 {\displaystyle \tau _{Y}}
 
 is replaced by a <a href="/facts/Comparison_of_topologies/5Q9F2fjf">coarser topology</a> and/or 
 
 
 
 
 τ
 
 X
 
 
 
 
 {\displaystyle \tau _{X}}
 
 is replaced by a <a href="/facts/Comparison_of_topologies/5Q9F2fjf">finer topology</a>.

<h3>Homeomorphisms</h3>
Symmetric to the concept of a continuous map is an <a href="/facts/Open_map/abHUKX7l">open map</a>, for which images of open sets are open. If an open map f has an <a href="/facts/Inverse_function/Tg5nnXlu">inverse function</a>, that inverse is continuous, and if a continuous map g has an inverse, that inverse is open. Given a <a href="/facts/Bijective/j4gTuTmW">bijective</a> function f between two topological spaces, the inverse function 
 
 
 
 
 f
 
 −
 1
 
 
 
 
 {\displaystyle f^{-1}}
 
 need not be continuous. A bijective continuous function with a continuous inverse function is called a <a href="/facts/Homeomorphism/jMfWg3Mn">homeomorphism</a>.
If a continuous bijection has as its <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> a <a href="/facts/Compact_space/d0cgXJH7">compact space</a> and its codomain is <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff</a>, then it is a homeomorphism.

<h3>Defining topologies via continuous functions</h3>
Given a function

f
        :
        X
        →
        S
        ,
      
    
    {\displaystyle f:X\to S,}

where X is a topological space and S is a set (without a specified topology), the <a href="/facts/Final_topology/gk2UV5va">final topology</a> on S is defined by letting the open sets of S be those subsets A of S for which 
 
 
 
 
 f
 
 −
 1
 
 
 (
 A
 )
 
 
 {\displaystyle f^{-1}(A)}
 
 is open in X. If S has an existing topology, f is continuous with respect to this topology if and only if the existing topology is <a href="/facts/Comparison_of_topologies/5Q9F2fjf">coarser</a> than the final topology on S. Thus, the final topology is the finest topology on S that makes f continuous. If f is <a href="/facts/Surjective/KezhLpCb">surjective</a>, this topology is canonically identified with the <a href="/facts/Quotient_topology/muE0qQVC">quotient topology</a> under the <a href="/facts/Equivalence_relation/1MQ5rtkW">equivalence relation</a> defined by f.
Dually, for a function f from a set S to a topological space X, the <a href="/facts/Initial_topology/bNRdJM81">initial topology</a> on S is defined by designating as an open set every subset A of S such that 
 
 
 
 A
 =
 
 f
 
 −
 1
 
 
 (
 U
 )
 
 
 {\displaystyle A=f^{-1}(U)}
 
 for some open subset U of X. If S has an existing topology, f is continuous with respect to this topology if and only if the existing topology is finer than the initial topology on S. Thus, the initial topology is the coarsest topology on S that makes f continuous. If f is injective, this topology is canonically identified with the <a href="/facts/Subspace_topology/NjU6z4Ca">subspace topology</a> of S, viewed as a subset of X.
A topology on a set S is uniquely determined by the class of all continuous functions 
 
 
 
 S
 →
 X
 
 
 {\displaystyle S\to X}
 
 into all topological spaces X. <a href="/facts/Duality_(mathematics)/8jGuyQIU">Dually</a>, a similar idea can be applied to maps 
 
 
 
 X
 →
 S
 .
 
 
 {\displaystyle X\to S.}

<h2 id="related-notions">Related notions</h2>
If 
 
 
 
 f
 :
 S
 →
 Y
 
 
 {\displaystyle f:S\to Y}
 
 is a continuous function from some subset 
 
 
 
 S
 
 
 {\displaystyle S}
 
 of a topological space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 then a continuous extension of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 to 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is any continuous function 
 
 
 
 F
 :
 X
 →
 Y
 
 
 {\displaystyle F:X\to Y}
 
 such that 
 
 
 
 F
 (
 s
 )
 =
 f
 (
 s
 )
 
 
 {\displaystyle F(s)=f(s)}
 
 for every 
 
 
 
 s
 ∈
 S
 ,
 
 
 {\displaystyle s\in S,}
 
 which is a condition that often written as 
 
 
 
 f
 =
 F
 
 
 
 |
 
 
 
 S
 
 
 .
 
 
 {\displaystyle f=F{\big \vert }_{S}.}
 
 In words, it is any continuous function 
 
 
 
 F
 :
 X
 →
 Y
 
 
 {\displaystyle F:X\to Y}
 
 that <a href="/facts/Restriction_of_a_function/JV59DhKy">restricts</a> to 
 
 
 
 f
 
 
 {\displaystyle f}
 
 on 
 
 
 
 S
 .
 
 
 {\displaystyle S.}
 
 This notion is used, for example, in the <a href="/facts/Tietze_extension_theorem/B7GJWFUL">Tietze extension theorem</a> and the <a href="/facts/Hahn%E2%80%93Banach_theorem/3AUHusvn">Hahn–Banach theorem</a>. If 
 
 
 
 f
 :
 S
 →
 Y
 
 
 {\displaystyle f:S\to Y}
 
 is not continuous, then it could not possibly have a continuous extension. If 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 is a <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff space</a> and 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is a <a href="/facts/Dense_set/nPT9Yxao">dense subset</a> of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 then a continuous extension of 
 
 
 
 f
 :
 S
 →
 Y
 
 
 {\displaystyle f:S\to Y}
 
 to 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 if one exists, will be unique. The <a href="/facts/Blumberg_theorem/h015Form">Blumberg theorem</a> states that if 
 
 
 
 f
 :
 
 R
 
 →
 
 R
 
 
 
 {\displaystyle f:\mathbb {R} \to \mathbb {R} }
 
 is an arbitrary function then there exists a dense subset 
 
 
 
 D
 
 
 {\displaystyle D}
 
 of 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
 such that the restriction 
 
 
 
 f
 
 
 
 |
 
 
 
 D
 
 
 :
 D
 →
 
 R
 
 
 
 {\displaystyle f{\big \vert }_{D}:D\to \mathbb {R} }
 
 is continuous; in other words, every function 
 
 
 
 
 R
 
 →
 
 R
 
 
 
 {\displaystyle \mathbb {R} \to \mathbb {R} }
 
 can be restricted to some dense subset on which it is continuous. 
Various other mathematical domains use the concept of continuity in different but related meanings. For example, in <a href="/facts/Order_theory/gn6fM3oJ">order theory</a>, an order-preserving function 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\to Y}
 
 between particular types of <a href="/facts/Partially_ordered_set/qb4a3FAX">partially ordered sets</a> 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 is continuous if for each <a href="/facts/Directed_set/GmWcqe3L">directed subset</a> 
 
 
 
 A
 
 
 {\displaystyle A}
 
 of 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 we have 
 
 
 
 sup
 f
 (
 A
 )
 =
 f
 (
 sup
 A
 )
 .
 
 
 {\displaystyle \sup f(A)=f(\sup A).}
 
 Here 
 
 
 
 
 sup
 
 
 
 {\displaystyle \,\sup \,}
 
 is the <a href="/facts/Supremum/oXCKVopz">supremum</a> with respect to the orderings in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 ,
 
 
 {\displaystyle Y,}
 
 respectively. This notion of continuity is the same as topological continuity when the partially ordered sets are given the <a href="/facts/Scott_topology/7CqoURj0">Scott topology</a>.<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a><a class="footnote-ref" id="fnref:21" href="#fn:21">21</a>
In <a href="/facts/Category_theory/6zS3DXPa">category theory</a>, a <a href="/facts/Functor/l2u3rIBE">functor</a>

F
        :
        
          
            C
          
        
        →
        
          
            D
          
        
      
    
    {\displaystyle F:{\mathcal {C}}\to {\mathcal {D}}}

between two <a href="/facts/Category_(mathematics)/7WzxUThf">categories</a> is called <a href="/facts/Continuous_functor/JoLugLs3">continuous</a> if it commutes with small <a href="/facts/Limit_(category_theory)/JoLugLs3">limits</a>. That is to say,

lim
              ←
            
          
          
            i
            ∈
            I
          
        
        ⁡
        F
        (
        
          C
          
            i
          
        
        )
        ≅
        F
        
          (
          
            
              
                
                  lim
                  ←
                
              
              
                i
                ∈
                I
              
            
            ⁡
            
              C
              
                i
              
            
          
          )
        
      
    
    {\displaystyle \varprojlim _{i\in I}F(C_{i})\cong F\left(\varprojlim _{i\in I}C_{i}\right)}

for any small (that is, indexed by a set 
 
 
 
 I
 ,
 
 
 {\displaystyle I,}
 
 as opposed to a <a href="/facts/Class_(mathematics)/F5EmDd9b">class</a>) <a href="/facts/Diagram_(category_theory)/kGTlM945">diagram</a> of <a href="/facts/Object_(category_theory)/7WzxUThf">objects</a> in 
 
 
 
 
 
 C
 
 
 
 
 {\displaystyle {\mathcal {C}}}
 
.
A <a href="/facts/Continuity_space/sKbl02pB">continuity space</a> is a generalization of metric spaces and posets,<a class="footnote-ref" id="fnref:22" href="#fn:22">22</a><a class="footnote-ref" id="fnref:23" href="#fn:23">23</a> which uses the concept of <a href="/facts/Quantale/j0Rell32">quantales</a>, and that can be used to unify the notions of metric spaces and <a href="/facts/Domain_theory/G1wRMK2K">domains</a>.<a class="footnote-ref" id="fnref:24" href="#fn:24">24</a>
In <a href="/facts/Measure_theory/yCq7zlI4">measure theory</a>, a function 
 
 
 
 f
 :
 E
 →
 
 
 R
 
 
 k
 
 
 
 
 {\displaystyle f:E\to \mathbb {R} ^{k}}
 
 defined on a <a href="/facts/Lebesgue_measurable_set/8us9cGoW">Lebesgue measurable set</a> 
 
 
 
 E
 ⊆
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle E\subseteq \mathbb {R} ^{n}}
 
 is called approximately continuous at a point 
 
 
 
 
 x
 
 0
 
 
 ∈
 E
 
 
 {\displaystyle x_{0}\in E}
 
 if the <a href="/facts/Approximate_limit/9Emh19qm">approximate limit</a> of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 at 
 
 
 
 
 x
 
 0
 
 
 
 
 {\displaystyle x_{0}}
 
 exists and equals 
 
 
 
 f
 (
 
 x
 
 0
 
 
 )
 
 
 {\displaystyle f(x_{0})}
 
. This generalizes the notion of continuity by replacing the ordinary limit with the <a href="/facts/Approximate_limit/9Emh19qm">approximate limit</a>. A fundamental result known as the Stepanov-Denjoy theorem states that a function is <a href="/facts/Measurable_function/NgIHnb1b">measurable</a> if and only if it is approximately continuous <a href="/facts/Almost_everywhere/1quRFtJE">almost everywhere</a>.<a class="footnote-ref" id="fnref:25" href="#fn:25">25</a>

<h2 id="see-also">See also</h2>

<ul><li><a href="/facts/Continuity_(mathematics)/n8CGmpos">Continuity (mathematics)</a></li>
<li><a href="/facts/Absolute_continuity/NPKDt9Qn">Absolute continuity</a></li>
<li>Approximate continuity</li>
<li><a href="/facts/Dini_continuity/u8swn7ES">Dini continuity</a></li>
<li><a href="/facts/Equicontinuity/cKjFQWD6">Equicontinuity</a></li>
<li><a href="/facts/Geometric_continuity/mffL4ch2">Geometric continuity</a></li>
<li><a href="/facts/Parametric_continuity/mffL4ch2">Parametric continuity</a></li>
<li><a href="/facts/Classification_of_discontinuities/TGLMsQom">Classification of discontinuities</a></li>
<li><a href="/facts/Coarse_function/G0VB0EHX">Coarse function</a></li>
<li><a href="/facts/Continuous_function_(set_theory)/oBwh40rV">Continuous function (set theory)</a></li>
<li><a href="/facts/Continuous_stochastic_process/GEmGTXYm">Continuous stochastic process</a></li>
<li><a href="/facts/Normal_function/GCXHAu3H">Normal function</a></li>
<li><a href="/facts/Open_and_closed_maps/abHUKX7l">Open and closed maps</a></li>
<li><a href="/facts/Piecewise/nlQwptpe">Piecewise</a></li>
<li><a href="/facts/Symmetrically_continuous_function/fuIRL3xR">Symmetrically continuous function</a></li></ul>

<ul><li><a href="/facts/Direction-preserving_function/RfSJEytS">Direction-preserving function</a> - an analog of a continuous function in discrete spaces.</li></ul>

Wikimedia Commons has media related to Continuity (functions).

<h2 id="bibliography">Bibliography</h2>
<ul><li><a href="/facts/James_Dugundji/xCy4DgGy">Dugundji, James</a> (1966). Topology. Boston: Allyn and Bacon. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-697-06889-7. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/395340485">395340485</a>.</li>
<li><a href="https://www.encyclopediaofmath.org/index.php?title=Continuous_function">"Continuous function"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Bolzano, Bernard (1817). "Rein analytischer Beweis des Lehrsatzes daß zwischen je zwey Werthen, die ein entgegengesetzetes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege". Prague: Haase. <a href="http://dml.cz/handle/10338.dmlcz/400352" target="_blank">http://dml.cz/handle/10338.dmlcz/400352</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Dugac, Pierre (1973), "Eléments d'Analyse de Karl Weierstrass", Archive for History of Exact Sciences, 10 (1–2): 41–176, doi:10.1007/bf00343406, S2CID 122843140 <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Goursat, E. (1904), A course in mathematical analysis, Boston: Ginn, p. 2 <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Jordan, M.C. (1893), Cours d'analyse de l'École polytechnique, vol. 1 (2nd ed.), Paris: Gauthier-Villars, p. 46 <a href="https://books.google.com/books?id=h2VKAAAAMAAJ&pg=PA46" target="_blank">https://books.google.com/books?id=h2VKAAAAMAAJ&pg=PA46</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Harper, J.F. (2016), "Defining continuity of real functions of real variables", BSHM Bulletin: Journal of the British Society for the History of Mathematics, 31 (3): 1–16, doi:10.1080/17498430.2015.1116053, S2CID 123997123 <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Rusnock, P.; Kerr-Lawson, A. (2005), "Bolzano and uniform continuity", Historia Mathematica, 32 (3): 303–311, doi:10.1016/j.hm.2004.11.003 <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Speck, Jared (2014). "Continuity and Discontinuity" (PDF). MIT Math. p. 3. Archived from the original (PDF) on 2016-10-06. Retrieved 2016-09-02. Example 5. The function 
 
 
 
 1
 
 /
 
 x
 
 
 {\displaystyle 1/x}
 
 is continuous on 
 
 
 
 (
 0
 ,
 ∞
 )
 
 
 {\displaystyle (0,\infty )}
 
 and on 
 
 
 
 (
 −
 ∞
 ,
 0
 )
 ,
 
 
 {\displaystyle (-\infty ,0),}
 
, i.e., for 
 
 
 
 x
 >
 0
 
 
 {\displaystyle x>0}
 
 and for 
 
 
 
 x
 <
 0
 ,
 
 
 {\displaystyle x<0,}
 
 in other words, at every point in its domain. However, it is not a continuous function since its domain is not an interval. It has a single point of discontinuity, namely 
 
 
 
 x
 =
 0
 ,
 
 
 {\displaystyle x=0,}
 
, and an infinite discontinuity there. <a href="https://web.archive.org/web/20161006014646/http://math.mit.edu/~jspeck/18.01_Fall%202014/Supplementary%20notes/01c.pdf" target="_blank">https://web.archive.org/web/20161006014646/http://math.mit.edu/~jspeck/18.01_Fall%202014/Supplementary%20notes/01c.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Lang, Serge (1997), Undergraduate analysis, Undergraduate Texts in Mathematics (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-94841-6, section II.4 <a href="978-0-387-94841-6" target="_blank">978-0-387-94841-6</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Introduction to Real Analysis, updated April 2010, William F. Trench, Theorem 3.5.2, p. 172 <a href="http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF" target="_blank">http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Introduction to Real Analysis, updated April 2010, William F. Trench, 3.5 "A More Advanced Look at the Existence of the Proper Riemann Integral", pp. 171–177 <a href="http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF" target="_blank">http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">"Elementary Calculus". wisc.edu. <a href="http://www.math.wisc.edu/~keisler/calc.html" target="_blank">http://www.math.wisc.edu/~keisler/calc.html</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Brown, James Ward (2009), Complex Variables and Applications (8th ed.), McGraw Hill, p. 54, ISBN 978-0-07-305194-9 <a href="978-0-07-305194-9" target="_blank">978-0-07-305194-9</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Gaal, Steven A. (2009), Point set topology, New York: Dover Publications, ISBN 978-0-486-47222-5, section IV.10 <a href="978-0-486-47222-5" target="_blank">978-0-486-47222-5</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Searcóid, Mícheál Ó (2006), Metric spaces, Springer undergraduate mathematics series, Berlin, New York: Springer-Verlag, ISBN 978-1-84628-369-7, section 9.4 <a href="978-1-84628-369-7" target="_blank">978-1-84628-369-7</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Dugundji 1966, pp. 211–221. - Dugundji, James (1966). Topology. Boston: Allyn and Bacon. ISBN 978-0-697-06889-7. OCLC 395340485. <a href="https://search.worldcat.org/oclc/395340485" target="_blank">https://search.worldcat.org/oclc/395340485</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Dugundji 1966, pp. 211–221. - Dugundji, James (1966). Topology. Boston: Allyn and Bacon. ISBN 978-0-697-06889-7. OCLC 395340485. <a href="https://search.worldcat.org/oclc/395340485" target="_blank">https://search.worldcat.org/oclc/395340485</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Shurman, Jerry (2016). Calculus and Analysis in Euclidean Space (illustrated ed.). Springer. pp. 271–272. ISBN 978-3-319-49314-5. <a href="978-3-319-49314-5" target="_blank">978-3-319-49314-5</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">"general topology - Continuity and interior". Mathematics Stack Exchange. <a href="https://math.stackexchange.com/q/1209229" target="_blank">https://math.stackexchange.com/q/1209229</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Dugundji 1966, pp. 211–221. - Dugundji, James (1966). Topology. Boston: Allyn and Bacon. ISBN 978-0-697-06889-7. OCLC 395340485. <a href="https://search.worldcat.org/oclc/395340485" target="_blank">https://search.worldcat.org/oclc/395340485</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">Goubault-Larrecq, Jean (2013). Non-Hausdorff Topology and Domain Theory: Selected Topics in Point-Set Topology. Cambridge University Press. ISBN 978-1107034136. <a href="978-1107034136" target="_blank">978-1107034136</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">Gierz, G.; Hofmann, K. H.; Keimel, K.; Lawson, J. D.; Mislove, M. W.; Scott, D. S. (2003). Continuous Lattices and Domains. Encyclopedia of Mathematics and its Applications. Vol. 93. Cambridge University Press. ISBN 0521803381. <a href="0521803381" target="_blank">0521803381</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">Flagg, R. C. (1997). "Quantales and continuity spaces". Algebra Universalis. 37 (3): 257–276. CiteSeerX 10.1.1.48.851. doi:10.1007/s000120050018. S2CID 17603865. <a href="/wiki/CiteSeerX_(identifier)" target="_blank">/wiki/CiteSeerX_(identifier)</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></li>
<li id="fn:23">Kopperman, R. (1988). "All topologies come from generalized metrics". American Mathematical Monthly. 95 (2): 89–97. doi:10.2307/2323060. JSTOR 2323060. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></li>
<li id="fn:24">Flagg, B.; Kopperman, R. (1997). "Continuity spaces: Reconciling domains and metric spaces". Theoretical Computer Science. 177 (1): 111–138. doi:10.1016/S0304-3975(97)00236-3. <a href="https://doi.org/10.1016%2FS0304-3975%2897%2900236-3" target="_blank">https://doi.org/10.1016%2FS0304-3975%2897%2900236-3</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></li>
<li id="fn:25">Federer, H. (1969). Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften. Vol. 153. New York: Springer-Verlag. <a href="#fnref:25" class="footnote-back-ref">↩</a></li>
</ol>

Continuous function open-in-new

Continuous function