Definition 3

One says that a vector function, defined on an interval $I$, with values in a complete normed space $E$ over $\mathbf{R}$, is a regulated function, if it is the uniform limit of step functions on every compact subset of $I$.

In other words, the regulated functions are the elements of the closure in $\mathscr{F}_c(I; E)$ of the subspace $\mathscr{E}$ of step functions; $\overline{\mathscr{E}}$ is a vector subspace of $\mathscr{F}_c(I; E)$ and since $\mathscr{F}_c(I; E)$ is complete, so is $\overline{\mathscr{E}}$; in other words, if a function is the uniform limit of regulated functions on every compact subset of $I$, then it is regulated on $I$. For $f$ to be regulated on an interval $I$ it is necessary and sufficient that its restriction to every compact interval contained in $I$ be regulated.

Cor. I to II, p. 53 shows: