Theorem 27.3.5 (Mean Value Theorem).label Let $E$ be a topological vector space over $K \in \RC$, $F$ be a separated locally convex space over $K$, $\sigma \subset \mathfrak{B}(E)$ be a covering ideal, $V \subset E$ be open and star shaped at $x \in V$, and $f: V \to F$ be $\tilde \sigma$-differentiable on $V$, then for any $y \in V$,
\[f(y) - f(x) \in \overline{\text{Conv}\bracs{Df(z)(y - x)|z \in (x, y)}}\]
where $[x, y] = \bracs{(1 - t)x + ty|y \in [0, 1]}$.
Proof. Let $g: [0, 1] \to F$ be defined by $g(t) = f((1 - t)x + ty)$, then $g$ is differentiable with $Dg(t) = Df((1 - t)x + ty)(y - x)$, and continuous by Proposition 27.9.1.
By the Mean Value Theorem, $f(y) - f(x) = g(1) - g(0)$ is contained in
\[\overline{\text{Conv}\bracs{Dg(t)|t \in [0, 1]}}= \overline{\text{Conv}\bracs{Df(z)(y - x)|z \in (x, y)}}\]
$\square$
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