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## Definition: Step Functions

Let $$[a,b]$$ be a closed real interval. A function $$f:[a,b]\mapsto\mathbb R$$ is called a step function over the interval $$[a,b]$$ (or a staircase function over this interval), if there exist real numbers $$x_i$$ such that

$a=x_0 < x_1 < \ldots < x_{n-1} < x_n=b$

and on any open interval $$(x_{i-1},x_{i})$$, $$i=1,\ldots,n$$, the function $$f$$ is constant. The numbers $$x_0,\ldots,x_n$$ are called a partition of the real interval $$[a,b]$$.

In the following figure, you can generate different partitions and different step functions for the interval $$[-9,9]$$:

| | | | | created: 2016-03-01 21:43:42 | modified: 2020-01-08 12:26:44 | by: bookofproofs

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