Exercise 3, § 1

Let $f$ be a regulated function on an interval $I \subset \mathbf{R}$, taking its values in a complete normed space $E$ over $\mathbf{R}$: show that for every compact subset $H$ of $I$ the set $f(H)$ is relatively compact in $E$; give an example where $f(H)$ is not closed in $E$.