Step function

<h2 id="definition-and-first-consequences">Definition and first consequences</h2>
<p>A function 
  
    
      
        f
        :
        
          R
        
        →
        
          R
        
      
    
    {\displaystyle f\colon \mathbb {R} \rightarrow \mathbb {R} }
  
 is called a step function if it can be written as 
</p>

f
        (
        x
        )
        =
        
          ∑
          
            i
            =
            0
          
          
            n
          
        
        
          α
          
            i
          
        
        
          χ
          
            
              A
              
                i
              
            
          
        
        (
        x
        )
      
    
    {\displaystyle f(x)=\sum \limits _{i=0}^{n}\alpha _{i}\chi _{A_{i}}(x)}
  
, for all real numbers 
  
    
      
        x
      
    
    {\displaystyle x}

<p>where 
  
    
      
        n
        ≥
        0
      
    
    {\displaystyle n\geq 0}
  
, 
  
    
      
        
          α
          
            i
          
        
      
    
    {\displaystyle \alpha _{i}}
  
 are real numbers, 
  
    
      
        
          A
          
            i
          
        
      
    
    {\displaystyle A_{i}}
  
 are intervals, and 
  
    
      
        
          χ
          
            A
          
        
      
    
    {\displaystyle \chi _{A}}
  
 is the <a href="/facts/Indicator_function/QEuh04NM">indicator function</a> of 
  
    
      
        A
      
    
    {\displaystyle A}
  
:
</p>

χ
          
            A
          
        
        (
        x
        )
        =
        
          
            {
            
              
                
                  1
                
                
                  
                    if 
                  
                  x
                  ∈
                  A
                
              
              
                
                  0
                
                
                  
                    if 
                  
                  x
                  ∉
                  A
                
              
            
            
          
        
      
    
    {\displaystyle \chi _{A}(x)={\begin{cases}1&{\text{if }}x\in A\\0&{\text{if }}x\notin A\\\end{cases}}}

<p>In this definition, the intervals 
  
    
      
        
          A
          
            i
          
        
      
    
    {\displaystyle A_{i}}
  
 can be assumed to have the following two properties: 
</p>
<ol><li>The intervals are <a href="/facts/Disjoint_sets/nLr8XRVd">pairwise disjoint</a>: 
  
    
      
        
          A
          
            i
          
        
        ∩
        
          A
          
            j
          
        
        =
        ∅
      
    
    {\displaystyle A_{i}\cap A_{j}=\emptyset }
  
 for 
  
    
      
        i
        ≠
        j
      
    
    {\displaystyle i\neq j}
  
</li>
<li>The <a href="/facts/Union_(set_theory)/KVVXKfWl">union</a> of the intervals is the entire real line: 
  
    
      
        
          ⋃
          
            i
            =
            0
          
          
            n
          
        
        
          A
          
            i
          
        
        =
        
          R
        
        .
      
    
    {\displaystyle \bigcup _{i=0}^{n}A_{i}=\mathbb {R} .}
  
</li></ol>
<p>Indeed, if that is not the case to start with, a different set of intervals can be picked for which these assumptions hold. For example, the step function
</p>

f
        =
        4
        
          χ
          
            [
            −
            5
            ,
            1
            )
          
        
        +
        3
        
          χ
          
            (
            0
            ,
            6
            )
          
        
      
    
    {\displaystyle f=4\chi _{[-5,1)}+3\chi _{(0,6)}}

<p>can be written as
</p>

f
        =
        0
        
          χ
          
            (
            −
            ∞
            ,
            −
            5
            )
          
        
        +
        4
        
          χ
          
            [
            −
            5
            ,
            0
            ]
          
        
        +
        7
        
          χ
          
            (
            0
            ,
            1
            )
          
        
        +
        3
        
          χ
          
            [
            1
            ,
            6
            )
          
        
        +
        0
        
          χ
          
            [
            6
            ,
            ∞
            )
          
        
        .
      
    
    {\displaystyle f=0\chi _{(-\infty ,-5)}+4\chi _{[-5,0]}+7\chi _{(0,1)}+3\chi _{[1,6)}+0\chi _{[6,\infty )}.}

<h3>Variations in the definition</h3>
<p>Sometimes, the intervals are required to be right-open<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> or allowed to be singleton.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> The condition that the collection of intervals must be finite is often dropped, especially in school mathematics,<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> though it must still be <a href="/facts/Locally_finite_collection/WTjQrJ1S">locally finite</a>, resulting in the definition of piecewise constant functions.
</p>
<h2 id="examples">Examples</h2>

<ul><li>A <a href="/facts/Constant_function/BmPx6dn7">constant function</a> is a trivial example of a step function. Then there is only one interval, 
  
    
      
        
          A
          
            0
          
        
        =
        
          R
        
        .
      
    
    {\displaystyle A_{0}=\mathbb {R} .}
  
</li>
<li>The <a href="/facts/Sign_function/VADy9zQq">sign function</a> sgn(<i>x</i>), which is −1 for negative numbers and +1 for positive numbers, and is the simplest non-constant step function.</li>
<li>The <a href="/facts/Heaviside_step_function/bgsUtxgP">Heaviside function</a> <i>H</i>(<i>x</i>), which is 0 for negative numbers and 1 for positive numbers, is equivalent to the sign function, up to a shift and scale of range (
  
    
      
        H
        =
        (
        sgn
        +
        1
        )
        
          /
        
        2
      
    
    {\displaystyle H=(\operatorname {sgn} +1)/2}
  
). It is the mathematical concept behind some test <a href="/facts/Signal_(electronics)/41YqUJ8c">signals</a>, such as those used to determine the <a href="/facts/Step_response/7pq4VyNt">step response</a> of a <a href="/facts/Dynamical_system_(definition)/hcaspdq4">dynamical system</a>.</li></ul>

<ul><li>The <a href="/facts/Rectangular_function/cVm8QG4F">rectangular function</a>, the normalized <a href="/facts/Boxcar_function/oPj0WQYJ">boxcar function</a>, is used to model a unit pulse.</li></ul>
<h3>Non-examples</h3>
<ul><li>The <a href="/facts/Integer_part/ssehyMMf">integer part</a> function is not a step function according to the definition of this article, since it has an infinite number of intervals. However, some authors<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> also define step functions with an infinite number of intervals.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a></li></ul>
<h2 id="properties">Properties</h2>
<ul><li>The sum and product of two step functions is again a step function. The product of a step function with a number is also a step function. As such, the step functions form an <a href="/facts/Algebra_over_a_field/8T8ulWJb">algebra</a> over the real numbers.</li>
<li>A step function takes only a finite number of values. If the intervals 
  
    
      
        
          A
          
            i
          
        
        ,
      
    
    {\displaystyle A_{i},}
  
 for 
  
    
      
        i
        =
        0
        ,
        1
        ,
        …
        ,
        n
      
    
    {\displaystyle i=0,1,\dots ,n}
  
 in the above definition of the step function are disjoint and their union is the real line, then 
  
    
      
        f
        (
        x
        )
        =
        
          α
          
            i
          
        
      
    
    {\displaystyle f(x)=\alpha _{i}}
  
 for all 
  
    
      
        x
        ∈
        
          A
          
            i
          
        
        .
      
    
    {\displaystyle x\in A_{i}.}
  
</li>
<li>The <a href="/facts/Definite_integral/V7lyV997">definite integral</a> of a step function is a <a href="/facts/Piecewise_linear_function/6bKlCbEa">piecewise linear function</a>.</li>
<li>The <a href="/facts/Lebesgue_integral/f4AgvC6x">Lebesgue integral</a> of a step function 
  
    
      
        
          f
          =
          
            ∑
            
              i
              =
              0
            
            
              n
            
          
          
            α
            
              i
            
          
          
            χ
            
              
                A
                
                  i
                
              
            
          
        
      
    
    {\displaystyle \textstyle f=\sum _{i=0}^{n}\alpha _{i}\chi _{A_{i}}}
  
 is 
  
    
      
        
          ∫
          f
          
          d
          x
          =
          
            ∑
            
              i
              =
              0
            
            
              n
            
          
          
            α
            
              i
            
          
          ℓ
          (
          
            A
            
              i
            
          
          )
          ,
        
      
    
    {\displaystyle \textstyle \int f\,dx=\sum _{i=0}^{n}\alpha _{i}\ell (A_{i}),}
  
 where 
  
    
      
        ℓ
        (
        A
        )
      
    
    {\displaystyle \ell (A)}
  
 is the length of the interval 
  
    
      
        A
      
    
    {\displaystyle A}
  
, and it is assumed here that all intervals 
  
    
      
        
          A
          
            i
          
        
      
    
    {\displaystyle A_{i}}
  
 have finite length. In fact, this equality (viewed as a definition) can be the first step in constructing the Lebesgue integral.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a></li>
<li>A <a href="/facts/Discrete_random_variable/TwTBXnLT">discrete random variable</a> is sometimes defined as a <a href="/facts/Random_variable/TwTBXnLT">random variable</a> whose <a href="/facts/Cumulative_distribution_function/WaKU8tp4">cumulative distribution function</a> is piecewise constant.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> In this case, it is locally a step function (globally, it may have an infinite number of steps). Usually however, any random variable with only countably many possible values is called a discrete random variable, in this case their cumulative distribution function is not necessarily locally a step function, as infinitely many intervals can accumulate in a finite region.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Crenel_function/PY4Px8Xk">Crenel function</a></li>
<li><a href="/facts/Piecewise/nlQwptpe">Piecewise</a></li>
<li><a href="/facts/Sigmoid_function/lAXu3AwW">Sigmoid function</a></li>
<li><a href="/facts/Simple_function/hWAJkmDQ">Simple function</a></li>
<li><a href="/facts/Step_detection/dUIACN1T">Step detection</a></li>
<li><a href="/facts/Heaviside_step_function/bgsUtxgP">Heaviside step function</a></li>
<li><a href="/facts/Piecewise-constant_valuation/yDUhIwSF">Piecewise-constant valuation</a></li></ul>

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

<ol>
<li id="fn:1"><p>"Step Function". <a href="http://mathworld.wolfram.com/StepFunction.html" target="_blank">http://mathworld.wolfram.com/StepFunction.html</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>"Step Functions - Mathonline". <a href="http://mathonline.wikidot.com/step-functions" target="_blank">http://mathonline.wikidot.com/step-functions</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>"Mathwords: Step Function". <a href="https://www.mathwords.com/s/step_function.htm" target="_blank">https://www.mathwords.com/s/step_function.htm</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>"Archived copy". Archived from the original on 2015-09-12. Retrieved 2024-12-16.{{cite web}}:  CS1 maint: archived copy as title (link) <a href="https://web.archive.org/web/20150912010951/http://study.com:80/academy/lesson/step-function-definition-equation-examples.html" target="_blank">https://web.archive.org/web/20150912010951/http://study.com:80/academy/lesson/step-function-definition-equation-examples.html</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>"Step Function". <a href="https://www.varsitytutors.com/hotmath/hotmath_help/topics/step-function" target="_blank">https://www.varsitytutors.com/hotmath/hotmath_help/topics/step-function</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Bachman, Narici, Beckenstein (5 April 2002). "Example 7.2.2". Fourier and Wavelet Analysis. Springer, New York, 2000. ISBN 0-387-98899-8.{{cite book}}:  CS1 maint: multiple names: authors list (link) <a href="0-387-98899-8" target="_blank">0-387-98899-8</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Bachman, Narici, Beckenstein (5 April 2002). "Example 7.2.2". Fourier and Wavelet Analysis. Springer, New York, 2000. ISBN 0-387-98899-8.{{cite book}}:  CS1 maint: multiple names: authors list (link) <a href="0-387-98899-8" target="_blank">0-387-98899-8</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>Weir, Alan J (10 May 1973). "3". Lebesgue integration and measure. Cambridge University Press, 1973. ISBN 0-521-09751-7. <a href="0-521-09751-7" target="_blank">0-521-09751-7</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>Bertsekas, Dimitri P. (2002). Introduction to Probability. Tsitsiklis, John N., Τσιτσικλής, Γιάννης Ν. Belmont, Mass.: Athena Scientific. ISBN 188652940X. OCLC 51441829. <a href="188652940X" target="_blank">188652940X</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
</ol>

Step function open-in-new

Step function