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 \([-9,9]\):

  1. Proposition: Step Functions as a Subspace of all Functions on a Closed Real Interval
  2. Definition: Order Relation for Step Functions

Definitions: 1 2
Lemmas: 3
Proofs: 4 5 6 7 8 9 10
Propositions: 11 12

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