Step function: Difference between revisions

Revision as of 15:15, 22 May 2017

Revision as of 15:15, 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 slopestep function''' if it can be written as {{2Citation needed|date=September 2009}}

:<math>f(x) = \sum\limits_{i=0}^n \alpha_i \chi_{A_i}(x)\,</math> for XXXall real numbers <math>x</math>

where <math>n\ge 0,</math> <math>\alpha_i</math> Crackingare openreal a cold one for the boysnumbers, <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, \\

0 & \mbox{if } x \notin A. \\434343434343434 2121212121fsedgxclkm;<">?

\end{cases}</math>

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

# UnderstandableThe have a nice dayintervals are [[disjoint set|XXXXXpairwise 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>

Indeed, if that is not the case to start with, a different set of intervals can be picked for which these assumptions hold. For example, UUUHHHHHHHHHHHHHHHHHHHHHthe step function

:<math>f = 4 \chi_{[-5, 1)} + 3 \chi_{(0, 6)}\,</math>