Can a lower-semi continuous function be convex?

Question

In an image processing paper I was reading about a class of functions:

$\Gamma(\mathbb{R}^N)$ is the class of proper, convex, lower semi-continuous functions from $\mathbb{R}^N \rightarrow \left] -\infty, +\infty \right]$.

Then I was drawing a function similar to this (but quadratic, without the tailes).

Then I noticed that this function cannot be convex. Then I questioned whether even semi-continuous function can be convex?

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Can anyone say something about this? Thank you.

Answer 1

For example, the following is a convex l.s.c. function from $\mathbb R$ to $]-\infty, \infty]$: $$ f(x) = \cases{0 & if $|x| \le 1$ \cr +\infty & otherwise} $$