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→Definition and first consequences: In the first figure, added more information about how the function in this figure is defined. |
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In this definition, the intervals <math>A_i</math> can be assumed to have the following two properties:
# The intervals are [[disjoint
# The [[union (set theory)|union]] of the intervals is the entire real line: <math>\bigcup_{i=0}^n A_i = \mathbb R.</math>
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===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
==Examples==
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* The [[rectangular function]], the normalized [[boxcar function]], is used to model a unit pulse.
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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<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=
==See also==
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