Definition:Step Function

Definition
A real function $f: \R \to \R$ is a step function 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:


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


 * $\mathbb I_1, \mathbb I_2, \ldots, \mathbb I_n$ 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

 * Definition:Heaviside Step Function