Exercise 14, § 1

Let $\mathbf{f}$ be a vector function which is the primitive of a regulated function $\mathbf{f}'$ on $[a, b]$, and such that $\mathbf{f}(a) = \mathbf{f}(b) = 0$. Show that if $M$ is the least upper bound of $\| \mathbf{f}'(x) \|$ over the set of points of $[a, b]$ where $\mathbf{f}'$ is continuous, then