Exercise 32, § 2

Let K be a compact convex set in a Hausdorff topological vector space E and let $f$ be an upper semi-continuous convex function defined over K. Show that $f$ is bounded over K. (Observe first that $f$ is bounded above in K; if $f$ is not bounded below show that $\liminf_{y \to x, y \neq x} f(y) = -\infty$ for every point $x \in K$, and that this contradicts Baire’s theorem.) Give an example where $f$ is not continuous.