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| {{About|a piecewise constant function|the unit step function|Heaviside step function}}
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| In [[mathematics]], a [[function (mathematics)|function]] on the [[real number]]s is called a '''step function''' (or '''staircase function''') if it can be written as a [[finite set|finite]] [[linear combination]] of [[indicator function]]s of [[interval (mathematics)|interval]]s. Informally speaking, a step function is a [[piecewise]] [[constant function]] having only finitely many pieces.
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| [[Image:StepFunctionExample.png|thumb|right|250px|Example of a step function (the red graph). This particular step function is [[Continuous_function#Directional_and_semi-continuity|right-continuous]].]]
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| ==Definition and first consequences==
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| A function <math>f: \mathbb{R} \rightarrow \mathbb{R}</math> is called a '''step function''' if it can be written as {{Citation needed|date=September 2009}}
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| :<math>f(x) = \sum\limits_{i=0}^n \alpha_i \chi_{A_i}(x)\,</math> for all real numbers <math>x</math>
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| where <math>n\ge 0,</math> <math>\alpha_i</math> are real numbers, <math>A_i</math> are intervals, and <math>\chi_A\,</math> (sometimes written as <math>1_A</math>) is the [[indicator function]] of <math>A</math>:
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| :<math>\chi_A(x) =
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| \begin{cases}
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| 1 & \mbox{if } x \in A, \\
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| 0 & \mbox{if } x \notin A. \\
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| \end{cases}
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| </math>
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| In this definition, the intervals <math>A_i</math> can be assumed to have the following two properties:
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| # The intervals are [[disjoint set|disjoint]], <math>A_i\cap A_j=\emptyset</math> for <math>i\ne j</math>
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| # The [[union (set theory)|union]] of the intervals is the entire real line, <math>\cup_{i=0}^n A_i=\mathbb R.</math>
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| 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
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| : <math>f = 4 \chi_{[-5, 1)} + 3 \chi_{(0, 6)}\,</math> | |
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| can be written as
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| : <math>f = 0\chi_{(-\infty, -5)} +4 \chi_{[-5, 0]} +7 \chi_{(0, 1)} + 3 \chi_{[1, 6)}+0\chi_{[6, \infty)}.\,</math>
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| ==Examples==
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| [[Image:Dirac distribution CDF.svg|325px|thumb|The [[Heaviside step function]] is an often used step function.]]
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| * A [[constant function]] is a trivial example of a step function. Then there is only one interval, <math>A_0=\mathbb R.</math>
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| * The [[Heaviside step function|Heaviside function]] ''H''(''x'') is an important step function. It is the mathematical concept behind some test [[Signal (electronics)|signals]], such as those used to determine the [[step response]] of a [[dynamical system (definition)|dynamical system]].
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| [[File:Rectangular function.svg|thumb|The [[rectangular function]], the next simplest step function.]]
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| * The [[rectangular function]], the normalized [[boxcar function]], is the next simplest step function, and is used to model a unit pulse.
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| === Non-examples ===
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| * The [[integer part]] function is not a step function according to the definition of this article, since it has an infinite number of intervals. However, some authors define step functions also with an infinite number of intervals.<ref>for example see: {{Cite book | author=Bachman, Narici, Beckenstein | title=Fourier and Wavelet Analysis | publisher=Springer, New York, 2000 | isbn=0-387-98899-8 | chapter =Example 7.2.2}}</ref>
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| ==Properties==
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| * 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 [[algebra over a field|algebra]] over the real numbers.
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| * A step function takes only a finite number of values. If the intervals <math>A_i,</math> <math>i=0, 1, \dots, n,</math> in the above definition of the step function are disjoint and their union is the real line, then <math>f(x)=\alpha_i\,</math> for all <math>x\in A_i.</math>
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| * The [[Lebesgue integral]] of a step function <math>\textstyle f = \sum\limits_{i=0}^n \alpha_i \chi_{A_i}\,</math> is <math>\textstyle \int \!f\,dx = \sum\limits_{i=0}^n \alpha_i \ell(A_i),\,</math> where <math>\ell(A)</math> is the length of the interval <math>A,</math> and it is assumed here that all intervals <math>A_i</math> have finite length. In fact, this equality (viewed as a definition) can be the first step in constructing the Lebesgue integral.<ref>{{Cite book | author=Weir, Alan J | authorlink= | coauthors= | title=Lebesgue integration and measure | date= | publisher=Cambridge University Press, 1973 | location= | isbn=0-521-09751-7 | unused_data=|chapter= 3}}</ref>
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| ==See also==
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| *[[Simple function]]
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| *[[Piecewise defined function]]
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| *[[Sigmoid function]]
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| *[[Step detection]]
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| ==References==
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| {{reflist}}
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| {{DEFAULTSORT:Step Function}}
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| [[Category:Special functions]]
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