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 \([-9,9]\):
Definitions: 1 2
Lemmas: 3
Proofs: 4 5 6 7 8 9 10
Propositions: 11 12