Let $\mathcal{X}$ be a real Hilbert space, and let $f\colon\mathcal{X}\to\mathbb{R}$ be convex.
It's known that if $\mathcal{X}$ is finite-dimensional, then $f$ is also continuous. However, if $\mathcal{X}$ is infinite dimensional, this result is no longer true, since there are discontinuous linear functionals (and linear functionals are convex).
I'd like to get a better intuition on this class of discontinuous, convex, real-valued functions, so I'm looking for more nonlinear examples. A simple affine example is constructed here. However, that example is not incredibly enlightening since it is based off of the linear case. Are there other examples?