Discontinuous convex function over an open convex set

Question

I’d like examples of discontinuous convex functions $f : \Omega \to \mathbb R$, where $\Omega$ is on open convex subset of a real vector space, $E$.

And the “more” discontinuous the better.

Answer 1

In a finite dimensional space convex functions on open sets are necessarily continuous. Rockafellar's book has a proof. In an infinite dimensional normed linear space there exist discontinuous linear functionals. These are very 'badly' discontinuous and very 'nicely' convex.