Definition 2

One says that a map $f$ of an interval $I \subset \mathbf{R}$ into a set E is a step function if there is a partition of I into a finite number of intervals $J_k$ such that $f$ is constant on each of the $J_k$.

Let $(a_i)_{0 \leq i \leq n}$ be the strictly increasing sequence formed by the distinct endpoints of the $J_k$; since the $J_k$ are pairwise disjoint each of them either reduces to a singleton $a_i$ or is an interval with two consecutive points $a_i, a_{i+1}$ as endpoints; moreover, since I is the union of the $J_k$, the point $a_0$ is the left-hand endpoint, and $a_n$ is the right-hand endpoint of I. Every step function on I can thus be characterised as a function which is constant on each of the open intervals ]$a_i,\ a_{i+1}[\ (0 \leq i \leq n - 1)$, where $(a_i)_{0 \leq i \leq n}$ is a strictly increasing sequence of points of $I$ with $a_0$ being the left-hand endpoint and $a_n$ the right-hand endpoint of $I$.