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{{Short description|Linear combination of indicator functions of real intervals}}

In [[mathematics]], a [[function (mathematics)|function]] on the [[real number]]s is called a '''step function''' if it can be written as a finite [[linear combination]] of [[indicator function]]s of [[half-open interval]]s. Informally speaking, a step function is a piecewise constant function having only finitely many pieces.

{{About|a piecewise constant function|the unit step function|Heaviside step function}}

In [[mathematics]], a [[function (mathematics)|function]] on the [[real number]]s is called a '''step function''' if it can be written as a [[finite set|finite]] [[linear combination]] of [[indicator function]]s of [[half-openinterval (mathematics)|interval]]s. Informally speaking, a step function is a [[piecewise]] [[constant function]] having only finitely many pieces.

[[Image:StepFunctionExample.png|thumb|right|250px|Example of a step function with n=4.]]

[[Image:StepFunctionExample.png|thumb|right|250px|An example of step functions (the red graph). In this function, each constant subfunction with a function value ''α_i'' (''i'' = 0, 1, 2, ...) is defined by an interval ''A_i'' and intervals are distinguished by points ''x_j'' (''j'' = 1, 2, ...). This particular step function is [[Continuous function#Directional and semi-continuity|right-continuous]].]]

Let the following quantities be given:

<ul>

<li> a [[sequence]] of coefficients

:<math>\{\alpha_0, \dots, \alpha_n\}\subset \mathbb{R},\; n \in \mathbb{N} \setminus \{0\}</math> </li>

<li> a sequence of interval margins

:<math>\{x_1 < \dots < x_{n-1}\} \subset \mathbb{R}</math>

</li>

<li> a sequence of intervals

:<math>A_0 := (-\infty, x_1)</math>

:<math>A_i := [x_i, x_{i+1})\,</math> (for <math>i=1,\cdots,n-2</math>)

:<math>A_n := [x_{n-1},\infty)</math>

(Although the intervals are shown as being closed below and open above, this is not necessary to the definition; all that is required is that the intervals A_n do not intersect, and that their union is the set of real numbers.)

</li>

==Definition and first consequences==

</ul>

'''Definition:''' Given the notations above, aA function <math>f:\colon \mathbb{R} \rightarrow \mathbb{R}</math> is called a '''step function''' [[if and only if]] it can be written as {{Citation needed|date=September 2009}}

:<math>f(x) = \sum\limits_{i=0}^n \alpha_i \cdot 1_chi_{A_i}(x)</math>, for all real numbers <math>x</math>

where </math>n\ge for all 0</math>x, <math>\inalpha_i</math> are real numbers, \mathbb{R}<math>A_i</math> whereare intervals, and <math>1_A\,chi_A</math> is the [[indicator function]] of <math>A\,</math>:

'''Definition:''' Given the notations above, a function <math>f: \mathbb{R} \rightarrow \mathbb{R}</math> is a '''step function''' [[if and only if]] it can be written as

:<math>\chi_A(x) = \begin{cases}

1, & \mathrmtext{if } \; x \in A \\

f(x) = \sum\limits_{i=0}^n \alpha_i \cdot 1_{A_i}(x)

0 & \text{if } x \notin A \\

</math> for all <math>x \in \mathbb{R}</math> where <math>1_A\,</math> is the [[indicator function]] of <math>A\,</math>:

: \end{cases}</math>1_A(x) =

\left\{

\begin{matrix}

1, & \mathrm{if} \; x \in A \\

0, & \mathrm{otherwise}.

\end{matrix}

\right.

</math>

In this definition, the intervals <math>A_i</math> can be assumed to have the following two properties:

'''Note:''' for all <math>i=0,\cdots,n</math> and <math>x \in A_i</math> it holds:

# The intervals are [[disjoint sets|pairwise disjoint]]: <math>A_i \cap A_j = \emptyset</math> for <math>i \neq j</math>

<math>f(x)=\alpha_i\,</math>.

# The [[union (set theory)|union]] of the intervals is the entire real line: <math>\bigcup_{i=0}^n A_i = \mathbb R.</math>

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

== Special step functions ==

:<math>A_nf := [x_4 \chi_{n[-5, 1)}, + 3 \inftychi_{(0, 6)}</math>

A particular step function, the ''unit step function'' or [[Heaviside step function]] ''H''(''x''), is obtained by setting ''n''=1, &alpha;₀=0, &alpha;₁=1, and ''x''₁=0 in the general expression above. It is the mathematical concept behind some test [[signal]]s, as those used to determine the [[step response]] of a [[dynamical system (definition)|dynamical system]].<!-- <math>n=1</math>, <math>\alpha_0=0</math>, <math>\alpha_1=1</math>, and <math>x_1=0</math>.-->

can be written as

:<math>f = 0\chi_{(-\infty, -5)} +4 \chi_{[-5, 0]} +7 \chi_{(0, 1)} + 3 \chi_{[1, 6)}+0\chi_{[6, \infty)}.</math>

===Variations in the definition===

Sometimes, the intervals are required to be right-open<ref>{{Cite web|url=http://mathworld.wolfram.com/StepFunction.html|title = Step Function}}</ref> or allowed to be singleton.<ref>{{Cite web|url=http://mathonline.wikidot.com/step-functions|title = Step Functions - Mathonline}}</ref> The condition that the collection of intervals must be finite is often dropped, especially in school mathematics,<ref>{{Cite web|url=https://www.mathwords.com/s/step_function.htm|title=Mathwords: Step Function}}</ref><ref>{{Cite web | title=Archived copy | url=https://study.com/academy/lesson/step-function-definition-equation-examples.html | archive-url=https://web.archive.org/web/20150912010951/http://study.com:80/academy/lesson/step-function-definition-equation-examples.html | access-date=2024-12-16 | archive-date=2015-09-12}}</ref><ref>{{Cite web|url=https://www.varsitytutors.com/hotmath/hotmath_help/topics/step-function|title = Step Function}}</ref> though it must still be [[Locally finite collection|locally finite]], resulting in the definition of piecewise constant functions.

==Examples==

[[Image:Dirac distribution CDF.svg|325px|thumb|The [[Heaviside step function]] is an often-used step function.]]

* A [[constant function]] is a trivial example of a step function. Then there is only one interval, <math>A_0=\mathbb R.</math>

* The [[sign function]] {{math|sgn(''x'')}}, which is −1 for negative numbers and +1 for positive numbers, and is the simplest non-constant step function.

* The [[Heaviside step function|Heaviside function]] {{math|''H''(''x'')}}, 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 (<math>H = (\sgn + 1)/2</math>). 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]].

[[File:Rectangular function.svg|thumb|The [[rectangular function]], the next simplest step function.]]

* The [[rectangular function]], the normalized [[boxcar function]], is used to model a unit pulse.

===Non-examples===

* 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<ref name=bachman_narici_beckenstein>{{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| date=5 April 2002 }}</ref> also define step functions with an infinite number of intervals.<ref name=bachman_narici_beckenstein />

==Properties==

* 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.

* A step function takes only a finite number of values. If the intervals <math>A_i,</math> for <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>

* The [[definite integral]] of a step function is a [[piecewise linear function]].

* The [[Lebesgue integral]] of a step function <math>\textstyle f = \sum_{i=0}^n \alpha_i \chi_{A_i}</math> is <math>\textstyle \int f\,dx = \sum_{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 | title=Lebesgue integration and measure | date= 10 May 1973| publisher=Cambridge University Press, 1973 | isbn=0-521-09751-7 |chapter= 3}}</ref>

* A [[discrete random variable]] is sometimes defined as a [[random variable]] whose [[cumulative distribution function]] is piecewise constant.<ref name=":0">{{Cite book|title=Introduction to Probability|last=Bertsekas|author-link=Dimitri Bertsekas|first=Dimitri P.|date=2002|publisher=Athena Scientific|others=[[John Tsitsiklis|Tsitsiklis, John N.]], Τσιτσικλής, Γιάννης Ν.|isbn=188652940X|___location=Belmont, Mass.|oclc=51441829}}</ref> 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.

==See also==

* [[simpleCrenel function]]

* [[Piecewise]]

*[[piecewise defined function]]

* [[Sigmoid function]]

* [[Simple function]]

* [[Step detection]]

* [[piecewiseHeaviside definedstep function]]

* [[Piecewise-constant valuation]]

==References==

[[Category:Elementary special functions]]

{{Reflist}}

{{DEFAULTSORT:Step Function}}

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