Editing Step function (section)

==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.