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

where <math>n\ge 0,</math> <math>\alpha_i</math> areCracking Xopen numbersa cold one for the boys, <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:

# TheUnderstandable intervalshave a nice day are [[disjoint set|XXXXX 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, the step functionUUUHHHHHHHHHHHHHHHHHHHHH

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