Exercise 30, § 2

a) Give an example of a convex function that is defined over a compact convex subset K of $\mathbf{R}^2$, that is bounded and lower semi-continuous in K, but is not continuous at a point of the frontier of K (consider the gauge of a disc of which 0 is a frontier point). b) Deduce from a) an example of a convex function defined in an open half-plane D of $\mathbf{R}^2$, not bounded above in D and not tending to a limit at a frontier point of D. c) Deduce from a) an example of a convex lower semi-continuous function defined over a compact convex subset A of $\mathbf{R}^2$ but not bounded above in A. (Take for A the set of points $(\xi, \eta)$ such that $\xi^4 \leq \eta \leq 1$ in $\mathbf{R}^2$.)