Editing Step function (section)

==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> is the [[indicator function]] of <math>A</math>:
:<math>\chi_A(x) = \begin{cases}
  1 & \text{if } x \in A \\
  0 & \text{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 sets|pairwise disjoint]]: <math>A_i \cap A_j = \emptyset</math> for <math>i \neq j</math>
# The [[union (set theory)|union]] of the intervals is the entire real line: <math>\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.