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.

Example of a step function (the red graph). This particular step function is right-continuous.

Definition and first consequences

A function ${\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} }$ 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] }}

${\displaystyle f(x)=\sum \limits _{i=0}^{n}\alpha _{i}\chi _{A_{i}}(x)\,}$ for all real numbers ${\displaystyle x}$
${\displaystyle \chi _{A}(x)={\begin{cases}1&{\mbox{if }}x\in A,\\0&{\mbox{if }}x\notin A.\\\end{cases}}}$

In this definition, the intervals ${\displaystyle A_{i}}$ can be assumed to have the following two properties:

1. The intervals are disjoint, ${\displaystyle A_{i}\cap A_{j}=\emptyset }$ for ${\displaystyle i\neq j}$
2. The union of the intervals is the entire real line, ${\displaystyle \cup _{i=0}^{n}A_{i}=\mathbb {R} .}$

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

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

can be written as

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

Examples

The Heaviside step function is an often used step function.
The rectangular function, the next simplest step function.

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.[1]