Theorem 18.3.4 (Mean Value Theorem). Let $-\infty < a < b < \infty$, $E$ be a separated locally convex space, $S \subset [a, b]$ be at most countable, and $f \in C([a, b]; E)$ be differentiable on $(a, b) \setminus N$, then
\[f(b) - f(a) \in \overline{\text{Conv}\bracs{Df(x)(b - a)| x \in (a, b) \setminus N}}\]
Proof. By Proposition 18.2.9, $f$ is right-differentiable on $(a, b) \setminus N$ with
\[D^{+}f(x) = Df(x) \in \bracs{Df(y)|y \in (a, b) \setminus N}\]
for all $x \in (a, b)$. Let $g(x) = x$, then by Theorem 18.3.3,
\[f(b) - f(a) \in \overline{(b - a)\text{Conv}\bracs{Df(x)|x \in (a, b) \setminus N}}\]
$\square$