In Rockafellar's convex analysis there was an example of improper convex function:
$$ f(x) = \begin{cases} -\infty & \text{if } ||x||<1, \\ 0 & \text{if } ||x|| =1, \\ +\infty & \text{if } ||x||>1 \end{cases} $$
How to verify that this is indeed a convex function?
The epigraph is $\operatorname{epi} f = (B(0,1)\times \mathbb{R}) \cup (\bar{B}(0,1) \times [0, \infty))$.
Suppose $x,y \in \operatorname{epi} f$. If both $x,y$ belong to either of the constituent sets, then it is clear that $[x,y]$ is also in the same set. So, suppose $x \in B(0,1)\times \mathbb{R}$ and $y \in (\bar{B}(0,1) \times [0, \infty)) \setminus (B(0,1)\times \mathbb{R})$. Since $\pi_1(\lambda x + (1-\lambda)y) \in B(0,1)$ for any $\lambda \in (0,1]$, we see that $(y,x] \subset B(0,1)\times \mathbb{R}$ and so $[x,y]$ is contained in the epigraph.
