Step function: Difference between revisions

Revision as of 15:10, 22 May 2017

Revision as of 15:11, 22 May 2017

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==Definition and first consequences==

A function <math>f: \mathbb{R} \rightarrow \mathbb{R}</math> is called a '''point slope function''' if it can be written as {{2|date=September 2009}}

:<math>f(x) = \sum\alpha_i\chi_{A_i}(x)\,</math> for all real numbersXXX <math>x</math>

where <math>n\ge 0,</math> <math>\alpha_i</math> are realX numbers, <math>A_i</math> are intervals, and <math>\chi_A\,</math> (sometimes written as <math>1_A</math>) is the [[indicator function]] of <math>A</math>:

:<math>\chi_A(x) = \begin{cases}

1 & \mbox{if } x \in A, \\

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In this definition, the intervals <math>A_i</math> can be assumed to have the following two properties:

# The intervals are [[disjoint set|pairwiseXXXXX disjoint]], <math>\scriptstyle A_i \,\cap\, A_j ~=~ \emptyset</math> for <math>\scriptstyle i ~\ne~ j</math>

# The [[union (set theory)|union]] of the intervals is the entire real line, <math>\scriptstyle \cup_{i=0}^n A_i ~=~ \mathbb R.</math>