Step function: Difference between revisions

Revision as of 17:53, 11 December 2020

Revision as of 14:51, 17 December 2020

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{{Use American English|date = March 2019}}

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

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

{{Use American English|date = March 2019}}

{{More footnotes|date=February 2019}}

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|Example of a step function (the red graph). This particular step function is [[Continuous_functionContinuous function#Directional_and_semiDirectional and semi-continuity|right-continuous]].]]

==Definition and first consequences==

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

* 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>\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 | 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=[[Dimitri_Bertsekas|Bertsekas]]|author-link=Dimitri_Bertsekas|first=Dimitri P.|date=2002|publisher=Athena Scientific|others=[[John_TsitsiklisJohn 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==

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==References==

{{reflistReflist}}

{{DEFAULTSORT:Step Function}}