Step function: Difference between revisions

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

*** THIS IS NOT HIGH SCHOOL MATERIAL***{{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''' (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.

[[Image:StepFunctionExample.png|thumb|right|250px|ExampleAn example of a step functionfunctions (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_functionContinuous function#Directional_and_semiDirectional and semi-continuity|right-continuous]].]]

==Definition and first consequences==

A function <math>f:\colon \mathbb{R} \rightarrow \mathbb{R}</math> is called a '''step function''' if it can be written as {{Citation needed|date=September 2009}}

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

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>:

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

1 & \mboxtext{if } x \in A, \\

0 & \mboxtext{if } x \notin A. \\

\end{cases}</math>

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

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

# The [[union (set theory)|union]] of the intervals is the entire real line,: <math>\scriptstyle \cup_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

:<math>f = 4 \chi_{[-5, 1)} + 3 \chi_{(0, 6)}\,</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'')</math>}}, 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 an important step function, and 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 the next simplest step function, and 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\limits_sum_{i=0}^n \alpha_i \chi_{A_i}\,</math> is <math>\textstyle \int \!f\,dx = \sum\limits_sum_{i=0}^n \alpha_i \ell(A_i),\,</math> where <math>\textstyle\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= 10 May 1973| publisher=Cambridge University Press, 1973 | ___location= | 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==

* [[Unit stepCrenel function]]

* [[Crenel functionPiecewise]]

* [[SimpleSigmoid function]]

* [[Piecewise definedSimple function]]

* [[SigmoidStep functiondetection]]

* [[Heaviside step function]]

*[[Step detection]]

* [[Piecewise-constant valuation]]

==References==

{{reflistReflist}}

{{DEFAULTSORT:Step Function}}