Exercise 14, § 2

Let $f$ be a vector function defined and continuous on a compact interval $I = [a, b]$ in $\mathbf{R}$, with values in a normed space over $\mathbf{R}$. Suppose that $f$ admits a right derivative at all points of the complement with respect to $[a, b[$ of a countable subset A of this interval. Show that there exists a point $x \in ]a, b[\cap \overline{A}$ such that

(Argue by contradiction, decomposing $[a, b[$ into three intervals $[a, t[$, $[t, t + h[$ and $[t + h, b[$ with $t \notin A$; if $k = \|f(b) - f(a)\|/(b - a)$, note that for $h$ sufficiently small one has $\|f(t + h) - f(t)\| < k.h$, and use th. 2 of I, p. 15 for the other intervals.)