Step function: Difference between revisions

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Revision as of 09:50, 1 April 2020

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=== 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==

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* 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_{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 | 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|url=https://www.worldcat.org/oclc/51441829|title=Introduction to Probability|last=[[Dimitri_Bertsekas|Bertsekas]]|first=Dimitri P.|date=2002|publisher=Athena Scientific|others=[[John_Tsitsiklis|Tsitsiklis, John N.,]] Τσιτσικλής, Γιάννης Ν.|year=|isbn=188652940X|___location=Belmont, Mass.|pages=|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==