Let the function $f: C \rightarrow \mathbb{R}$ be convex and continuous, where $C \subset \mathbb{R}^n$ is a compact set.
Prove or disprove that $f$ is Lipschitz continuous on $C$.
Comments: If $f$ were defined on an open set $O$, then I can show that it is Lipschitz continuous on every compact subset of $O$, see Proving that a convex function is Lipschitz for the scalar case. In this case, however, I need to exploit the assumption that $f$ is continuous also on the boundary of $C$ - continuity of $f$ in the interior of $C$ already follows from the convexity assumption.