# Definition:Step Function

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## Definition

A real function $f: \R \to \R$ is a **step function** if and only if it can be expressed as a finite linear combination of the form:

- $\map f x = \lambda_1 \chi_{\mathbb I_1} + \lambda_2 \chi_{\mathbb I_2} + \cdots + \lambda_n \chi_{\mathbb I_n}$

where:

- $\mathbb I_1, \mathbb I_2, \ldots, \mathbb I_n$ are open intervals, where these intervals partition $\R$ (except for the endpoints)

- $\chi_{\mathbb I_1}, \chi_{\mathbb I_2}, \ldots, \chi_{\mathbb I_n}$ are characteristic functions of $\mathbb I_1, \mathbb I_2, \ldots, \mathbb I_n$.

## Also see

## Sources

- 1961: I.N. Sneddon:
*Fourier Series*... (previous) ... (next): Chapter One: $\S 2$. Fourier Series

- Weisstein, Eric W. "Step Function." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/StepFunction.html

*This article incorporates material from simple functions on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*