Classification of discontinuities

<h2 id="classification">Classification</h2>
For each of the following, consider a <a href="/facts/Real-valued_function/jetqlJpK">real valued function</a> 
 
 
 
 f
 
 
 {\displaystyle f}
 
 of a real variable 
 
 
 
 x
 ,
 
 
 {\displaystyle x,}
 
 defined in a neighborhood of the point 
 
 
 
 
 x
 
 0
 
 
 
 
 {\displaystyle x_{0}}
 
 at which 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is discontinuous.

<h3>Removable discontinuity</h3>

Consider the <a href="/facts/Piecewise/nlQwptpe">piecewise</a> function

f
 (
 x
 )
 =
 
 
 {
 
 
 
 
 x
 
 2
 
 
 
 
 
  for 
 
 x
 <
 1
 
 
 
 
 0
 
 
 
  for 
 
 x
 =
 1
 
 
 
 
 2
 −
 x
 
 
 
  for 
 
 x
 >
 1
 
 
 
 
 
 
 
 
 {\displaystyle f(x)={\begin{cases}x^{2}&{\text{ for }}x<1\\0&{\text{ for }}x=1\\2-x&{\text{ for }}x>1\end{cases}}}

The point 
 
 
 
 
 x
 
 0
 
 
 =
 1
 
 
 {\displaystyle x_{0}=1}
 
 is a removable <a href="/facts/Continuous_function/sKbl02pB">discontinuity</a>. For this kind of discontinuity:
The <a href="/facts/One-sided_limit/zLfr8gc5">one-sided limit</a> from the negative direction:

L
          
            −
          
        
        =
        
          lim
          
            x
            →
            
              x
              
                0
              
              
                −
              
            
          
        
        f
        (
        x
        )
      
    
    {\displaystyle L^{-}=\lim _{x\to x_{0}^{-}}f(x)}

and the one-sided limit from the positive direction:

L
          
            +
          
        
        =
        
          lim
          
            x
            →
            
              x
              
                0
              
              
                +
              
            
          
        
        f
        (
        x
        )
      
    
    {\displaystyle L^{+}=\lim _{x\to x_{0}^{+}}f(x)}

at 
 
 
 
 
 x
 
 0
 
 
 
 
 {\displaystyle x_{0}}
 
 both exist, are finite, and are equal to 
 
 
 
 L
 =
 
 L
 
 −
 
 
 =
 
 L
 
 +
 
 
 .
 
 
 {\displaystyle L=L^{-}=L^{+}.}
 
 In other words, since the two one-sided limits exist and are equal, the limit 
 
 
 
 L
 
 
 {\displaystyle L}
 
 of 
 
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(x)}
 
 as 
 
 
 
 x
 
 
 {\displaystyle x}
 
 approaches 
 
 
 
 
 x
 
 0
 
 
 
 
 {\displaystyle x_{0}}
 
 exists and is equal to this same value. If the actual value of 
 
 
 
 f
 
 (
 
 x
 
 0
 
 
 )
 
 
 
 {\displaystyle f\left(x_{0}\right)}
 
 is not equal to 
 
 
 
 L
 ,
 
 
 {\displaystyle L,}
 
 then 
 
 
 
 
 x
 
 0
 
 
 
 
 {\displaystyle x_{0}}
 
 is called a removable discontinuity. This discontinuity can be removed to make 
 
 
 
 f
 
 
 {\displaystyle f}
 
 continuous at 
 
 
 
 
 x
 
 0
 
 
 ,
 
 
 {\displaystyle x_{0},}
 
 or more precisely, the function

g
        (
        x
        )
        =
        
          
            {
            
              
                
                  f
                  (
                  x
                  )
                
                
                  x
                  ≠
                  
                    x
                    
                      0
                    
                  
                
              
              
                
                  L
                
                
                  x
                  =
                  
                    x
                    
                      0
                    
                  
                
              
            
            
          
        
      
    
    {\displaystyle g(x)={\begin{cases}f(x)&x\neq x_{0}\\L&x=x_{0}\end{cases}}}

is continuous at 
  
    
      
        x
        =
        
          x
          
            0
          
        
        .
      
    
    {\displaystyle x=x_{0}.}

The term removable discontinuity is sometimes broadened to include a <a href="/facts/Removable_singularity/6MPFDNha">removable singularity</a>, in which the limits in both directions exist and are equal, while the function is <a href="/facts/Defined_and_undefined/Bc1Q3px4">undefined</a> at the point 
 
 
 
 
 x
 
 0
 
 
 .
 
 
 {\displaystyle x_{0}.}
 
<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> This use is an <a href="/facts/Abuse_of_terminology/D4W6n85F">abuse of terminology</a> because <a href="/facts/Continuous_function/sKbl02pB">continuity</a> and discontinuity of a function are concepts defined only for points in the function's domain.

<h3>Jump discontinuity</h3>

Consider the function

f
 (
 x
 )
 =
 
 
 {
 
 
 
 
 x
 
 2
 
 
 
 
 
 
  for 
 
 
 x
 <
 1
 
 
 
 
 0
 
 
 
 
  for 
 
 
 x
 =
 1
 
 
 
 
 2
 −
 (
 x
 −
 1
 
 )
 
 2
 
 
 
 
 
 
  for 
 
 
 x
 >
 1
 
 
 
 
 
 
 
 
 {\displaystyle f(x)={\begin{cases}x^{2}&{\mbox{ for }}x<1\\0&{\mbox{ for }}x=1\\2-(x-1)^{2}&{\mbox{ for }}x>1\end{cases}}}

Then, the point 
 
 
 
 
 x
 
 0
 
 
 =
 1
 
 
 {\displaystyle x_{0}=1}
 
 is a jump discontinuity.
In this case, a single limit does not exist because the one-sided limits, 
 
 
 
 
 L
 
 −
 
 
 
 
 {\displaystyle L^{-}}
 
 and 
 
 
 
 
 L
 
 +
 
 
 
 
 {\displaystyle L^{+}}
 
 exist and are finite, but are not equal: since, 
 
 
 
 
 L
 
 −
 
 
 ≠
 
 L
 
 +
 
 
 ,
 
 
 {\displaystyle L^{-}\neq L^{+},}
 
 the limit 
 
 
 
 L
 
 
 {\displaystyle L}
 
 does not exist. Then, 
 
 
 
 
 x
 
 0
 
 
 
 
 {\displaystyle x_{0}}
 
 is called a jump discontinuity, step discontinuity, or discontinuity of the first kind. For this type of discontinuity, the function 
 
 
 
 f
 
 
 {\displaystyle f}
 
 may have any value at 
 
 
 
 
 x
 
 0
 
 
 .
 
 
 {\displaystyle x_{0}.}

<h3>Essential discontinuity</h3>

For an essential discontinuity, at least one of the two one-sided limits does not exist in 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
. (Notice that one or both one-sided limits can be 
 
 
 
 ±
 ∞
 
 
 {\displaystyle \pm \infty }
 
).
Consider the function

f
 (
 x
 )
 =
 
 
 {
 
 
 
 sin
 ⁡
 
 
 5
 
 x
 −
 1
 
 
 
 
 
 
  for 
 
 x
 <
 1
 
 
 
 
 0
 
 
 
  for 
 
 x
 =
 1
 
 
 
 
 
 
 1
 
 x
 −
 1
 
 
 
 
 
 
  for 
 
 x
 >
 1.
 
 
 
 
 
 
 
 
 {\displaystyle f(x)={\begin{cases}\sin {\frac {5}{x-1}}&{\text{ for }}x<1\\0&{\text{ for }}x=1\\{\frac {1}{x-1}}&{\text{ for }}x>1.\end{cases}}}

Then, the point 
 
 
 
 
 x
 
 0
 
 
 =
 1
 
 
 {\displaystyle x_{0}=1}
 
 is an essential discontinuity.
In this example, both 
 
 
 
 
 L
 
 −
 
 
 
 
 {\displaystyle L^{-}}
 
 and 
 
 
 
 
 L
 
 +
 
 
 
 
 {\displaystyle L^{+}}
 
 do not exist in 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
, thus satisfying the condition of essential discontinuity. So 
 
 
 
 
 x
 
 0
 
 
 
 
 {\displaystyle x_{0}}
 
 is an essential discontinuity, infinite discontinuity, or discontinuity of the second kind. (This is distinct from an <a href="/facts/Essential_singularity/KMb1R292">essential singularity</a>, which is often used when studying <a href="/facts/Complex_analysis/hgCoQlH0">functions of complex variables</a>).

<h2 id="counting-discontinuities-of-a-function">Counting discontinuities of a function</h2>
Supposing that 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is a function defined on an interval 
 
 
 
 I
 ⊆
 
 R
 
 ,
 
 
 {\displaystyle I\subseteq \mathbb {R} ,}
 
 we will denote by 
 
 
 
 D
 
 
 {\displaystyle D}
 
 the set of all discontinuities of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 on 
 
 
 
 I
 .
 
 
 {\displaystyle I.}
 
 By 
 
 
 
 R
 
 
 {\displaystyle R}
 
 we will mean the set of all 
 
 
 
 
 x
 
 0
 
 
 ∈
 I
 
 
 {\displaystyle x_{0}\in I}
 
 such that 
 
 
 
 f
 
 
 {\displaystyle f}
 
 has a removable discontinuity at 
 
 
 
 
 x
 
 0
 
 
 .
 
 
 {\displaystyle x_{0}.}
 
 Analogously by 
 
 
 
 J
 
 
 {\displaystyle J}
 
 we denote the set constituted by all 
 
 
 
 
 x
 
 0
 
 
 ∈
 I
 
 
 {\displaystyle x_{0}\in I}
 
 such that 
 
 
 
 f
 
 
 {\displaystyle f}
 
 has a jump discontinuity at 
 
 
 
 
 x
 
 0
 
 
 .
 
 
 {\displaystyle x_{0}.}
 
 The set of all 
 
 
 
 
 x
 
 0
 
 
 ∈
 I
 
 
 {\displaystyle x_{0}\in I}
 
 such that 
 
 
 
 f
 
 
 {\displaystyle f}
 
 has an essential discontinuity at 
 
 
 
 
 x
 
 0
 
 
 
 
 {\displaystyle x_{0}}
 
 will be denoted by 
 
 
 
 E
 .
 
 
 {\displaystyle E.}
 
 Of course then 
 
 
 
 D
 =
 R
 ∪
 J
 ∪
 E
 .
 
 
 {\displaystyle D=R\cup J\cup E.}

The two following properties of the set 
 
 
 
 D
 
 
 {\displaystyle D}
 
 are relevant in the literature.

<ul><li>The set of 
 
 
 
 D
 
 
 {\displaystyle D}
 
 is an <a href="/facts/F-sigma_set/vI6Y6fdQ">
 
 
 
 
 F
 
 σ
 
 
 
 
 {\displaystyle F_{\sigma }}
 
 set</a>. The set of points at which a function is continuous is always a <a href="/facts/G-delta_set/Erg1BCFk">
 
 
 
 
 G
 
 δ
 
 
 
 
 {\displaystyle G_{\delta }}
 
 set</a> (see<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>).</li>
<li>If on the interval 
 
 
 
 I
 ,
 
 
 {\displaystyle I,}
 
 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is monotone then 
 
 
 
 D
 
 
 {\displaystyle D}
 
 is <a href="/facts/Countable/Wj4DmWPv">at most countable</a> and 
 
 
 
 D
 =
 J
 .
 
 
 {\displaystyle D=J.}
 
 This is <a href="/facts/Froda%2527s_theorem/P4eVKwIt">Froda's theorem</a>.</li></ul>
Tom Apostol<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> follows partially the classification above by considering only removable and jump discontinuities. His objective is to study the discontinuities of monotone functions, mainly to prove Froda’s theorem. With the same purpose, Walter Rudin<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> and Karl R. Stromberg<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> study also removable and jump discontinuities by using different terminologies. However, furtherly, both authors state that 
 
 
 
 R
 ∪
 J
 
 
 {\displaystyle R\cup J}
 
 is always a countable set (see<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a><a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>).
The term essential discontinuity has evidence of use in mathematical context as early as 1889.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> However, the earliest use of the term alongside a mathematical definition seems to have been given in the work by John Klippert.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> Therein, Klippert also classified essential discontinuities themselves by subdividing the set 
 
 
 
 E
 
 
 {\displaystyle E}
 
 into the three following sets:

 
 
 
 
 E
 
 1
 
 
 =
 
 {
 
 
 x
 
 0
 
 
 ∈
 I
 :
 
 lim
 
 x
 →
 
 x
 
 0
 
 
 −
 
 
 
 
 f
 (
 x
 )
 
  and 
 
 
 lim
 
 x
 →
 
 x
 
 0
 
 
 +
 
 
 
 
 f
 (
 x
 )
 
  do not exist in 
 
 
 R
 
 
 }
 
 ,
 
 
 {\displaystyle E_{1}=\left\{x_{0}\in I:\lim _{x\to x_{0}^{-}}f(x){\text{ and }}\lim _{x\to x_{0}^{+}}f(x){\text{ do not exist in }}\mathbb {R} \right\},}

E
          
            2
          
        
        =
        
          {
          
            
              x
              
                0
              
            
            ∈
            I
            :
             
            
              lim
              
                x
                →
                
                  x
                  
                    0
                  
                  
                    −
                  
                
              
            
            f
            (
            x
            )
            
               exists in 
            
            
              R
            
            
               and 
            
            
              lim
              
                x
                →
                
                  x
                  
                    0
                  
                  
                    +
                  
                
              
            
            f
            (
            x
            )
            
               does not exist in 
            
            
              R
            
          
          }
        
        ,
      
    
    {\displaystyle E_{2}=\left\{x_{0}\in I:\ \lim _{x\to x_{0}^{-}}f(x){\text{ exists in }}\mathbb {R} {\text{ and }}\lim _{x\to x_{0}^{+}}f(x){\text{ does not exist in }}\mathbb {R} \right\},}

E
          
            3
          
        
        =
        
          {
          
            
              x
              
                0
              
            
            ∈
            I
            :
             
            
              lim
              
                x
                →
                
                  x
                  
                    0
                  
                  
                    −
                  
                
              
            
            f
            (
            x
            )
            
               does not exist in 
            
            
              R
            
            
               and 
            
            
              lim
              
                x
                →
                
                  x
                  
                    0
                  
                  
                    +
                  
                
              
            
            f
            (
            x
            )
            
               exists in 
            
            
              R
            
          
          }
        
        .
      
    
    {\displaystyle E_{3}=\left\{x_{0}\in I:\ \lim _{x\to x_{0}^{-}}f(x){\text{ does not exist in }}\mathbb {R} {\text{ and }}\lim _{x\to x_{0}^{+}}f(x){\text{ exists in }}\mathbb {R} \right\}.}

Of course 
 
 
 
 E
 =
 
 E
 
 1
 
 
 ∪
 
 E
 
 2
 
 
 ∪
 
 E
 
 3
 
 
 .
 
 
 {\displaystyle E=E_{1}\cup E_{2}\cup E_{3}.}
 
 Whenever 
 
 
 
 
 x
 
 0
 
 
 ∈
 
 E
 
 1
 
 
 ,
 
 
 {\displaystyle x_{0}\in E_{1},}
 
 
 
 
 
 
 x
 
 0
 
 
 
 
 {\displaystyle x_{0}}
 
 is called an essential discontinuity of first kind. Any 
 
 
 
 
 x
 
 0
 
 
 ∈
 
 E
 
 2
 
 
 ∪
 
 E
 
 3
 
 
 
 
 {\displaystyle x_{0}\in E_{2}\cup E_{3}}
 
 is said an essential discontinuity of second kind. Hence he enlarges the set 
 
 
 
 R
 ∪
 J
 
 
 {\displaystyle R\cup J}
 
 without losing its characteristic of being countable, by stating the following:

<ul><li>The set 
 
 
 
 R
 ∪
 J
 ∪
 
 E
 
 2
 
 
 ∪
 
 E
 
 3
 
 
 
 
 {\displaystyle R\cup J\cup E_{2}\cup E_{3}}
 
 is countable.</li></ul>
<h2 id="rewriting-lebesgues-theorem">Rewriting Lebesgue's theorem</h2>
When 
 
 
 
 I
 =
 [
 a
 ,
 b
 ]
 
 
 {\displaystyle I=[a,b]}
 
 and 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is a bounded function, it is well-known of the importance of the set 
 
 
 
 D
 
 
 {\displaystyle D}
 
 in the regard of the Riemann <a href="/facts/Riemann_integral/vWGrPhVh">integrability</a> of 
 
 
 
 f
 .
 
 
 {\displaystyle f.}
 
 In fact, <a href="/facts/Riemann_integral/vWGrPhVh">Lebesgue's theorem (also named Lebesgue-Vitali)</a> theorem) states that 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is Riemann integrable on 
 
 
 
 I
 =
 [
 a
 ,
 b
 ]
 
 
 {\displaystyle I=[a,b]}
 
 if and only if 
 
 
 
 D
 
 
 {\displaystyle D}
 
 is a set with Lebesgue's measure zero.
In this theorem seems that all type of discontinuities have the same weight on the obstruction that a bounded function 
 
 
 
 f
 
 
 {\displaystyle f}
 
 be Riemann integrable on 
 
 
 
 [
 a
 ,
 b
 ]
 .
 
 
 {\displaystyle [a,b].}
 
 Since countable sets are sets of Lebesgue's measure zero and a countable union of sets with Lebesgue's measure zero is still a set of Lebesgue's mesure zero, we are seeing now that this is not the case. In fact, the discontinuities in the set 
 
 
 
 R
 ∪
 J
 ∪
 
 E
 
 2
 
 
 ∪
 
 E
 
 3
 
 
 
 
 {\displaystyle R\cup J\cup E_{2}\cup E_{3}}
 
 are absolutely neutral in the regard of the Riemann integrability of 
 
 
 
 f
 .
 
 
 {\displaystyle f.}
 
 The main discontinuities for that purpose are the essential discontinuities of first kind and consequently the Lebesgue-Vitali theorem can be rewritten as follows:

<ul><li>A bounded function, 
 
 
 
 f
 ,
 
 
 {\displaystyle f,}
 
 is Riemann integrable on 
 
 
 
 [
 a
 ,
 b
 ]
 
 
 {\displaystyle [a,b]}
 
 if and only if the correspondent set 
 
 
 
 
 E
 
 1
 
 
 
 
 {\displaystyle E_{1}}
 
 of all essential discontinuities of first kind of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 has Lebesgue's measure zero.</li></ul>
The case where 
 
 
 
 
 E
 
 1
 
 
 =
 ∅
 
 
 {\displaystyle E_{1}=\varnothing }
 
 correspond to the following well-known classical complementary situations of Riemann integrability of a bounded function 
 
 
 
 f
 :
 [
 a
 ,
 b
 ]
 →
 
 R
 
 
 
 {\displaystyle f:[a,b]\to \mathbb {R} }
 
:

<ul><li>If 
 
 
 
 f
 
 
 {\displaystyle f}
 
 has right-hand limit at each point of 
 
 
 
 [
 a
 ,
 b
 [
 
 
 {\displaystyle [a,b[}
 
 then 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is Riemann integrable on 
 
 
 
 [
 a
 ,
 b
 ]
 
 
 {\displaystyle [a,b]}
 
 (see<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>)</li>
<li>If 
 
 
 
 f
 
 
 {\displaystyle f}
 
 has left-hand limit at each point of 
 
 
 
 ]
 a
 ,
 b
 ]
 
 
 {\displaystyle ]a,b]}
 
 then 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is Riemann integrable on 
 
 
 
 [
 a
 ,
 b
 ]
 .
 
 
 {\displaystyle [a,b].}
 
</li>
<li>If 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is a <a href="/facts/Regulated_function/yEew23kQ">regulated function</a> on 
 
 
 
 [
 a
 ,
 b
 ]
 
 
 {\displaystyle [a,b]}
 
 then 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is <a href="/facts/Regulated_function/yEew23kQ">Riemann integrable</a> on 
 
 
 
 [
 a
 ,
 b
 ]
 .
 
 
 {\displaystyle [a,b].}
 
</li></ul>
<h3>Examples</h3>
<a href="/facts/Thomae%2527s_function/SjCt2ahN">Thomae's function</a> is discontinuous at every non-zero <a href="/facts/Rational_point/j5v4cxuE">rational point</a>, but continuous at every <a href="/facts/Irrational_number/Psv19TET">irrational</a> point. One easily sees that those discontinuities are all removable. By the first paragraph, there does not exist a function that is continuous at every <a href="/facts/Rational_number/VJQmyY05">rational</a> point, but discontinuous at every irrational point.
The <a href="/facts/Indicator_function/QEuh04NM">indicator function</a> of the rationals, also known as the <a href="/facts/Dirichlet_function/d9o708HR">Dirichlet function</a>, is <a href="/facts/Nowhere_continuous/XbxlDzx1">discontinuous everywhere</a>. These discontinuities are all essential of the first kind too.
Consider now the ternary <a href="/facts/Cantor_set/6NUcMJvN">Cantor set</a> 
 
 
 
 
 
 C
 
 
 ⊂
 [
 0
 ,
 1
 ]
 
 
 {\displaystyle {\mathcal {C}}\subset [0,1]}
 
 and its indicator (or characteristic) function

1
 
 
 
 C
 
 
 
 (
 x
 )
 =
 
 
 {
 
 
 
 1
 
 
 x
 ∈
 
 
 C
 
 
 
 
 
 
 0
 
 
 x
 ∈
 [
 0
 ,
 1
 ]
 ∖
 
 
 C
 
 
 .
 
 
 
 
 
 
 
 
 {\displaystyle \mathbf {1} _{\mathcal {C}}(x)={\begin{cases}1&x\in {\mathcal {C}}\\0&x\in [0,1]\setminus {\mathcal {C}}.\end{cases}}}
 
 
<a href="/facts/Cantor_set/6NUcMJvN">One way to construct</a> the Cantor set 
 
 
 
 
 
 C
 
 
 
 
 {\displaystyle {\mathcal {C}}}
 
 is given by 
 
 
 
 
 
 C
 
 
 :=
 
 ⋂
 
 n
 =
 0
 
 
 ∞
 
 
 
 C
 
 n
 
 
 
 
 {\textstyle {\mathcal {C}}:=\bigcap _{n=0}^{\infty }C_{n}}
 
 where the sets 
 
 
 
 
 C
 
 n
 
 
 
 
 {\displaystyle C_{n}}
 
 are obtained by recurrence according to

C
          
            n
          
        
        =
        
          
            
              C
              
                n
                −
                1
              
            
            3
          
        
        ∪
        
          (
          
            
              
                2
                3
              
            
            +
            
              
                
                  C
                  
                    n
                    −
                    1
                  
                
                3
              
            
          
          )
        
        
           for 
        
        n
        ≥
        1
        ,
        
           and 
        
        
          C
          
            0
          
        
        =
        [
        0
        ,
        1
        ]
        .
      
    
    {\displaystyle C_{n}={\frac {C_{n-1}}{3}}\cup \left({\frac {2}{3}}+{\frac {C_{n-1}}{3}}\right){\text{ for }}n\geq 1,{\text{ and }}C_{0}=[0,1].}

In view of the discontinuities of the function 
 
 
 
 
 
 1
 
 
 
 C
 
 
 
 (
 x
 )
 ,
 
 
 {\displaystyle \mathbf {1} _{\mathcal {C}}(x),}
 
 let's assume a point 
 
 
 
 
 x
 
 0
 
 
 ∉
 
 
 C
 
 
 .
 
 
 {\displaystyle x_{0}\not \in {\mathcal {C}}.}

Therefore there exists a set 
 
 
 
 
 C
 
 n
 
 
 ,
 
 
 {\displaystyle C_{n},}
 
 used in the <a href="/facts/Cantor_set/6NUcMJvN">formulation of 
 
 
 
 
 
 C
 
 
 
 
 {\displaystyle {\mathcal {C}}}
 
</a>, which does not contain 
 
 
 
 
 x
 
 0
 
 
 .
 
 
 {\displaystyle x_{0}.}
 
 That is, 
 
 
 
 
 x
 
 0
 
 
 
 
 {\displaystyle x_{0}}
 
 belongs to one of the open intervals which were removed in the construction of 
 
 
 
 
 C
 
 n
 
 
 .
 
 
 {\displaystyle C_{n}.}
 
 This way, 
 
 
 
 
 x
 
 0
 
 
 
 
 {\displaystyle x_{0}}
 
 has a neighbourhood with no points of 
 
 
 
 
 
 C
 
 
 .
 
 
 {\displaystyle {\mathcal {C}}.}
 
 (In another way, the same conclusion follows taking into account that <a href="/facts/Cantor_set/6NUcMJvN">
 
 
 
 
 
 C
 
 
 
 
 {\displaystyle {\mathcal {C}}}
 
</a> is a closed set and so its complementary with respect to 
 
 
 
 [
 0
 ,
 1
 ]
 
 
 {\displaystyle [0,1]}
 
 is open). Therefore 
 
 
 
 
 
 1
 
 
 
 C
 
 
 
 
 
 {\displaystyle \mathbf {1} _{\mathcal {C}}}
 
 only assumes the value zero in some neighbourhood of 
 
 
 
 
 x
 
 0
 
 
 .
 
 
 {\displaystyle x_{0}.}
 
 Hence 
 
 
 
 
 
 1
 
 
 
 C
 
 
 
 
 
 {\displaystyle \mathbf {1} _{\mathcal {C}}}
 
 is continuous at 
 
 
 
 
 x
 
 0
 
 
 .
 
 
 {\displaystyle x_{0}.}

This means that the set 
 
 
 
 D
 
 
 {\displaystyle D}
 
 of all discontinuities of 
 
 
 
 
 
 1
 
 
 
 C
 
 
 
 
 
 {\displaystyle \mathbf {1} _{\mathcal {C}}}
 
 on the interval 
 
 
 
 [
 0
 ,
 1
 ]
 
 
 {\displaystyle [0,1]}
 
 is a subset of 
 
 
 
 
 
 C
 
 
 .
 
 
 {\displaystyle {\mathcal {C}}.}
 
 Since 
 
 
 
 
 
 C
 
 
 
 
 {\displaystyle {\mathcal {C}}}
 
 is an <a href="/facts/Cantor_set/6NUcMJvN">uncountable set with null Lebesgue measure</a>, also 
 
 
 
 D
 
 
 {\displaystyle D}
 
 is a null Lebesgue measure set and so in the regard of <a href="/facts/Riemann_integral/vWGrPhVh">Lebesgue-Vitali theorem</a> 
 
 
 
 
 
 1
 
 
 
 C
 
 
 
 
 
 {\displaystyle \mathbf {1} _{\mathcal {C}}}
 
 is a Riemann integrable function.
More precisely one has 
 
 
 
 D
 =
 
 
 C
 
 
 .
 
 
 {\displaystyle D={\mathcal {C}}.}
 
 In fact, since 
 
 
 
 
 
 C
 
 
 
 
 {\displaystyle {\mathcal {C}}}
 
 is a nonwhere dense set, if 
 
 
 
 
 x
 
 0
 
 
 ∈
 
 
 C
 
 
 
 
 {\displaystyle x_{0}\in {\mathcal {C}}}
 
 then no <a href="/facts/Neighbourhood_(mathematics)/XHKgrpkp">neighbourhood</a> 
 
 
 
 
 (
 
 
 x
 
 0
 
 
 −
 ε
 ,
 
 x
 
 0
 
 
 +
 ε
 
 )
 
 
 
 {\displaystyle \left(x_{0}-\varepsilon ,x_{0}+\varepsilon \right)}
 
 of 
 
 
 
 
 x
 
 0
 
 
 ,
 
 
 {\displaystyle x_{0},}
 
 can be contained in 
 
 
 
 
 
 C
 
 
 .
 
 
 {\displaystyle {\mathcal {C}}.}
 
 This way, any neighbourhood of 
 
 
 
 
 x
 
 0
 
 
 ∈
 
 
 C
 
 
 
 
 {\displaystyle x_{0}\in {\mathcal {C}}}
 
 contains points of 
 
 
 
 
 
 C
 
 
 
 
 {\displaystyle {\mathcal {C}}}
 
 and points which are not of 
 
 
 
 
 
 C
 
 
 .
 
 
 {\displaystyle {\mathcal {C}}.}
 
 In terms of the function 
 
 
 
 
 
 1
 
 
 
 C
 
 
 
 
 
 {\displaystyle \mathbf {1} _{\mathcal {C}}}
 
 this means that both 
 
 
 
 
 lim
 
 x
 →
 
 x
 
 0
 
 
 −
 
 
 
 
 
 
 1
 
 
 
 C
 
 
 
 (
 x
 )
 
 
 {\textstyle \lim _{x\to x_{0}^{-}}\mathbf {1} _{\mathcal {C}}(x)}
 
 and 
 
 
 
 
 lim
 
 x
 →
 
 x
 
 0
 
 
 +
 
 
 
 
 
 1
 
 
 C
 
 
 
 (
 x
 )
 
 
 {\textstyle \lim _{x\to x_{0}^{+}}1_{\mathcal {C}}(x)}
 
 do not exist. That is, 
 
 
 
 D
 =
 
 E
 
 1
 
 
 ,
 
 
 {\displaystyle D=E_{1},}
 
 where by 
 
 
 
 
 E
 
 1
 
 
 ,
 
 
 {\displaystyle E_{1},}
 
 as before, we denote the set of all essential discontinuities of first kind of the function 
 
 
 
 
 
 1
 
 
 
 C
 
 
 
 .
 
 
 {\displaystyle \mathbf {1} _{\mathcal {C}}.}
 
 Clearly 
 
 
 
 
 ∫
 
 0
 
 
 1
 
 
 
 
 1
 
 
 
 C
 
 
 
 (
 x
 )
 d
 x
 =
 0.
 
 
 {\textstyle \int _{0}^{1}\mathbf {1} _{\mathcal {C}}(x)dx=0.}

<h2 id="discontinuities-of-derivatives">Discontinuities of derivatives</h2>
Let 
 
 
 
 I
 ⊆
 
 R
 
 
 
 {\displaystyle I\subseteq \mathbb {R} }
 
 an open interval, let 
 
 
 
 F
 :
 I
 →
 
 R
 
 
 
 {\displaystyle F:I\to \mathbb {R} }
 
 be differentiable on 
 
 
 
 I
 ,
 
 
 {\displaystyle I,}
 
 and let 
 
 
 
 f
 :
 I
 →
 
 R
 
 
 
 {\displaystyle f:I\to \mathbb {R} }
 
 be the derivative of 
 
 
 
 F
 .
 
 
 {\displaystyle F.}
 
 That is, 
 
 
 
 
 F
 ′
 
 (
 x
 )
 =
 f
 (
 x
 )
 
 
 {\displaystyle F'(x)=f(x)}
 
 for every 
 
 
 
 x
 ∈
 I
 
 
 {\displaystyle x\in I}
 
.
According to <a href="/facts/Darboux%2527s_theorem_(analysis)/uYXRzISn">Darboux's theorem</a>, the derivative function 
 
 
 
 f
 :
 I
 →
 
 R
 
 
 
 {\displaystyle f:I\to \mathbb {R} }
 
 satisfies the intermediate value property.
The function 
 
 
 
 f
 
 
 {\displaystyle f}
 
 can, of course, be continuous on the interval 
 
 
 
 I
 ,
 
 
 {\displaystyle I,}
 
 in which case <a href="/facts/Intermediate_value_theorem/SaFdYE7W">Bolzano's theorem</a> also applies. Recall that Bolzano's theorem asserts that every continuous function satisfies the intermediate value property.
On the other hand, the converse is false: Darboux's theorem does not assume 
 
 
 
 f
 
 
 {\displaystyle f}
 
 to be continuous and the intermediate value property does not imply 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is continuous on 
 
 
 
 I
 .
 
 
 {\displaystyle I.}

Darboux's theorem does, however, have an immediate consequence on the type of discontinuities that 
 
 
 
 f
 
 
 {\displaystyle f}
 
 can have. In fact, if 
 
 
 
 
 x
 
 0
 
 
 ∈
 I
 
 
 {\displaystyle x_{0}\in I}
 
 is a point of discontinuity of 
 
 
 
 f
 
 
 {\displaystyle f}
 
, then necessarily 
 
 
 
 
 x
 
 0
 
 
 
 
 {\displaystyle x_{0}}
 
 is an essential discontinuity of 
 
 
 
 f
 
 
 {\displaystyle f}
 
.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>
This means in particular that the following two situations cannot occur:

<ol><li>
 
 
 
 
 x
 
 0
 
 
 
 
 {\displaystyle x_{0}}
 
 is a removable discontinuity of 
 
 
 
 f
 
 
 {\displaystyle f}
 
.</li><li>
 
 
 
 
 x
 
 0
 
 
 
 
 {\displaystyle x_{0}}
 
 is a jump discontinuity of 
 
 
 
 f
 
 
 {\displaystyle f}
 
.</li></ol>
Furthermore, two other situations have to be excluded (see John Klippert<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a>):

<ol><li>
 
 
 
 
 lim
 
 x
 →
 
 x
 
 0
 
 
 −
 
 
 
 
 f
 (
 x
 )
 =
 ±
 ∞
 .
 
 
 {\displaystyle \lim _{x\to x_{0}^{-}}f(x)=\pm \infty .}
 
</li><li>
 
 
 
 
 lim
 
 x
 →
 
 x
 
 0
 
 
 +
 
 
 
 
 f
 (
 x
 )
 =
 ±
 ∞
 .
 
 
 {\displaystyle \lim _{x\to x_{0}^{+}}f(x)=\pm \infty .}
 
</li></ol>
Observe that whenever one of the conditions (i), (ii), (iii), or (iv) is fulfilled for some 
 
 
 
 
 x
 
 0
 
 
 ∈
 I
 
 
 {\displaystyle x_{0}\in I}
 
 one can conclude that 
 
 
 
 f
 
 
 {\displaystyle f}
 
 fails to possess an antiderivative, 
 
 
 
 F
 
 
 {\displaystyle F}
 
, on the interval 
 
 
 
 I
 
 
 {\displaystyle I}
 
.
On the other hand, a new type of discontinuity with respect to any function 
 
 
 
 f
 :
 I
 →
 
 R
 
 
 
 {\displaystyle f:I\to \mathbb {R} }
 
 can be introduced: an essential discontinuity, 
 
 
 
 
 x
 
 0
 
 
 ∈
 I
 
 
 {\displaystyle x_{0}\in I}
 
, of the function 
 
 
 
 f
 
 
 {\displaystyle f}
 
, is said to be a fundamental essential discontinuity of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 if

 
 
 
 
 lim
 
 x
 →
 
 x
 
 0
 
 
 −
 
 
 
 
 f
 (
 x
 )
 ≠
 ±
 ∞
 
 
 {\displaystyle \lim _{x\to x_{0}^{-}}f(x)\neq \pm \infty }
 
 and 
 
 
 
 
 lim
 
 x
 →
 
 x
 
 0
 
 
 +
 
 
 
 
 f
 (
 x
 )
 ≠
 ±
 ∞
 .
 
 
 {\displaystyle \lim _{x\to x_{0}^{+}}f(x)\neq \pm \infty .}

Therefore if 
 
 
 
 
 x
 
 0
 
 
 ∈
 I
 
 
 {\displaystyle x_{0}\in I}
 
 is a discontinuity of a derivative function 
 
 
 
 f
 :
 I
 →
 
 R
 
 
 
 {\displaystyle f:I\to \mathbb {R} }
 
, then necessarily 
 
 
 
 
 x
 
 0
 
 
 
 
 {\displaystyle x_{0}}
 
 is a fundamental essential discontinuity of 
 
 
 
 f
 
 
 {\displaystyle f}
 
.
Notice also that when 
 
 
 
 I
 =
 [
 a
 ,
 b
 ]
 
 
 {\displaystyle I=[a,b]}
 
 and 
 
 
 
 f
 :
 I
 →
 
 R
 
 
 
 {\displaystyle f:I\to \mathbb {R} }
 
 is a bounded function, as in the assumptions of Lebesgue's theorem, we have for all 
 
 
 
 
 x
 
 0
 
 
 ∈
 (
 a
 ,
 b
 )
 
 
 {\displaystyle x_{0}\in (a,b)}
 
:

lim
          
            x
            →
            
              x
              
                0
              
              
                ±
              
            
          
        
        f
        (
        x
        )
        ≠
        ±
        ∞
        ,
      
    
    {\displaystyle \lim _{x\to x_{0}^{\pm }}f(x)\neq \pm \infty ,}

lim
          
            x
            →
            
              a
              
                +
              
            
          
        
        f
        (
        x
        )
        ≠
        ±
        ∞
        ,
      
    
    {\displaystyle \lim _{x\to a^{+}}f(x)\neq \pm \infty ,}
  
 and

lim
          
            x
            →
            
              b
              
                −
              
            
          
        
        f
        (
        x
        )
        ≠
        ±
        ∞
        .
      
    
    {\displaystyle \lim _{x\to b^{-}}f(x)\neq \pm \infty .}

Therefore any essential discontinuity of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is a fundamental one.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Removable_singularity/6MPFDNha">Removable singularity</a> – Undefined point on a holomorphic function which can be made regular</li>
<li><a href="/facts/Mathematical_singularity/Y5sMQY4I">Mathematical singularity</a> – Point where a function, a curve or another mathematical object does not behave regularlyPages displaying short descriptions of redirect targets</li>
<li><a href="/facts/Regular_space/gLvRIhRH">Extension by continuity</a> – Property of topological space</li>
<li><a href="/facts/Smoothness/mffL4ch2">Smoothness</a> – Number of derivatives of a function (mathematics)
<ul><li><a href="/facts/Geometric_continuity/mffL4ch2">Geometric continuity</a> – Number of derivatives of a function (mathematics)Pages displaying short descriptions of redirect targets</li>
<li><a href="/facts/Parametric_continuity/mffL4ch2">Parametric continuity</a> – Number of derivatives of a function (mathematics)Pages displaying short descriptions of redirect targets</li></ul></li></ul>
<h2 id="notes">Notes</h2>

<h2 id="sources">Sources</h2>
<ul><li>Malik, S.C.; Arora, Savita (1992). Mathematical Analysis (2nd ed.). New York: Wiley. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-470-21858-4.{{cite book}}: CS1 maint: publisher location (link)</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="https://planetmath.org/Discontinuous">"Discontinuous"</a>. <a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a>.</li>
<li><a href="http://demonstrations.wolfram.com/Discontinuity/">"Discontinuity"</a> by <a href="/facts/Ed_Pegg%2c_Jr./cvnLFn0A">Ed Pegg, Jr.</a>, <a href="/facts/The_Wolfram_Demonstrations_Project/l0sh4gln">The Wolfram Demonstrations Project</a>, 2007.</li>
<li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/Discontinuity.html">"Discontinuity"</a>. <a href="/facts/MathWorld/yc1jodbq">MathWorld</a>.</li>
<li>Kudryavtsev, L.D. (2001) [1994]. <a href="https://www.encyclopediaofmath.org/index.php?title=Discontinuity_point&oldid=12112">"Discontinuity point"</a>. <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>. <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>.</li></ul>

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

<ol>
<li id="fn:1">See, for example, the last sentence in the definition given at Mathwords.[1] <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Stromberg, Karl R. (2015). An Introduction to Classical Real Analysis. American Mathematical Society. p. 120. Ex. 3 (c). ISBN 978-1-4704-2544-9. <a href="978-1-4704-2544-9" target="_blank">978-1-4704-2544-9</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Apostol, Tom (1974). Mathematical Analysis (2nd ed.). Addison and Wesley. p. 92, sec. 4.22, sec. 4.23 and Ex. 4.63. ISBN 0-201-00288-4. <a href="0-201-00288-4" target="_blank">0-201-00288-4</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Walter, Rudin (1976). Principles of Mathematical Analysis (third ed.). McGraw-Hill. pp. 94, Def. 4.26, Thms. 4.29 and 4.30. ISBN 0-07-085613-3. <a href="0-07-085613-3" target="_blank">0-07-085613-3</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Stromberg, Karl R. Op. cit. p. 128, Def. 3.87, Thm. 3.90. <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Walter, Rudin. Op. cit. p. 100, Ex. 17. <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Stromberg, Karl R. Op. cit. p. 131, Ex. 3. <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Whitney, William Dwight (1889). The Century Dictionary: An Encyclopedic Lexicon of the English Language. Vol. 2. London and New York: T. Fisher Unwin and The Century Company. p. 1652. ISBN 9781334153952. Archived from the original on 2008-12-16. An essential discontinuity is a discontinuity in which the value of the function becomes entirely indeterminable. {{cite book}}: ISBN / Date incompatibility (help) <a href="9781334153952" target="_blank">9781334153952</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Klippert, John (February 1989). "Advanced Advanced Calculus: Counting the Discontinuities of a Real-Valued Function with Interval Domain". Mathematics Magazine. 62: 43–48. doi:10.1080/0025570X.1989.11977410. <a href="http://about.jstor.org/terms" target="_blank">http://about.jstor.org/terms</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Metzler, R. C. (1971). "On Riemann Integrability". American Mathematical Monthly. 78 (10): 1129–1131. doi:10.1080/00029890.1971.11992961. <a href="https://doi.org/10.1080/00029890.1971.11992961" target="_blank">https://doi.org/10.1080/00029890.1971.11992961</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Rudin, Walter. Op.cit. pp. 109, Corollary. <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Klippert, John (2000). "On a discontinuity of a derivative". International Journal of Mathematical Education in Science and Technology. 31:S2: 282–287. doi:10.1080/00207390050032252. <a href="https://dx.doi.org/10.1080/00207390050032252" target="_blank">https://dx.doi.org/10.1080/00207390050032252</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
</ol>

Classification of discontinuities open-in-new

Classification of discontinuities