Nonlinear discontinuous convex function

Question

Let $X$ be a normed vector space. Construct nonlinear discontinuous convex function $f:X \rightarrow \mathbb{R}$.

I tried something with $\frac{-1}{\|x\|}$ but had no success.

Answer 1

Such example does not always exist. For example, if $\dim X<\infty$, $X$ is isomorphic to $\mathbb{R}^n$ and every convex function $f:\mathbb{R}^n\to \mathbb{R}$ is continuous.

For $\dim X=\infty$, recall that every infinite-dimensional normed space contains a discontinuous linear functional, say $f$. Let $g=f+c$, with constant $c\not=0$.

Then $g$ is convex and discontinuous but not linear, since $g(0)\not=0$.