Definition 11.5.2 (Step Map). Let $[a, b] \subset \real$, $E$ be a TVS, $f: [a, b] \to E$ be a function, and $P = \bracsn{x_j}_{1}^{n} \in \scp([a, b])$, then $f$ is a step map with respect to $P$ if for each $1 \le j \le n$, $f$ is constant on $(x_{j - 1}, x_{j})$.