Definition:Step Function

Definition
Let $f: \R \to \R$ be a real function.

Then $f$ is said to be a step function if it can be written as a finite linear combination of the form:


 * $f \left({x}\right) = \lambda_1 \chi_{\mathbb I_1} + \lambda_2 \chi_{\mathbb I_2} + \cdots + \lambda_n \chi_{\mathbb I_n}$

where:


 * $\lambda_1, \lambda_2, \cdots, \lambda_n$ are real constants


 * $\mathbb I_1, \mathbb I_2, \cdots, \mathbb I_n$ are intervals, where these intervals partition $\R$


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

Also see

 * Definition:Heaviside Step Function