Exercise 14, § 1

Let $\mathbf{f}$ be a vector-valued primitive function of a regulated function $\mathbf{f}'$ in $[a, b]$, and such that $\mathbf{f}(a) = \mathbf{f}(b) = 0$. Now prove that if $M$ is the supremum of $\| \mathbf{f}'(x) \|$ in the set of points of $[a, b]$ where $\mathbf{f}'$ is continuous, one has