# Step function

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In mathematics, a function on the real numbers is called a step function (or staircase function) if it can be written as a finite linear combination of indicator functions of intervals. Informally speaking, a step function is a piecewise constant function having only finitely many pieces.

## Definition and first consequences

A function $f:\mathbb {R} \rightarrow \mathbb {R}$ is called a step function if it can be written as {{ safesubst:#invoke:Unsubst||date=__DATE__ |\$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}

$f(x)=\sum \limits _{i=0}^{n}\alpha _{i}\chi _{A_{i}}(x)\,$ for all real numbers $x$ $\chi _{A}(x)={\begin{cases}1&{\mbox{if }}x\in A,\\0&{\mbox{if }}x\notin A.\\\end{cases}}$ In this definition, the intervals $A_{i}$ can be assumed to have the following two properties:

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

$f=4\chi _{[-5,1)}+3\chi _{(0,6)}\,$ can be written as

$f=0\chi _{(-\infty ,-5)}+4\chi _{[-5,0]}+7\chi _{(0,1)}+3\chi _{[1,6)}+0\chi _{[6,\infty )}.\,$ ## Examples

### 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 define step functions also with an infinite number of intervals.