Category:Definitions/Step Functions

This category contains definitions related to Step Functions.
Related results can be found in Category:Step Functions.

A step function is a real function whose graph consists entirely of unconnected horizontal line segments.

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:

$\ds \map f x$

$=$

$\ds \sum_{i \mathop = 1}^n \lambda_i \chi_{\mathbb I_i}$

$\ds $

$=$

$\ds \lambda_1 \chi_{\mathbb I_1} + \lambda_2 \chi_{\mathbb I_2} + \cdots + \lambda_n \chi_{\mathbb I_n}$

where:

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

$\mathbb I_1, \mathbb I_2, \ldots, \mathbb I_n$ are half-open intervals which partition $\R$

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

Subcategories

This category has only the following subcategory.

The following 5 pages are in this category, out of 5 total.