Let $f:\mathbb{R}^n\rightarrow \mathbb{R}\cup \{\pm \infty\}$ and let $S(f):=\{x ∈R^n | f(x) < +∞\}$ (effective domain)
How can I prove that $S(f)$ is convex doesn't imply that $f$ is convex?
Attempt: I'm thinking of showing that $S(f)$ can vary depending whether $f(x)$ is a proper or improper function?