A Function on a Convex Subset of $\mathbb{R}^n$ Is Convex If and Only If Its Epigraph Is Convex

In the setting of Convex Functions on $\mathbb{R}^n$: Standing Notation, let $C\subseteq\mathbb{R}^{n}$ be convex and let $f:C\to\mathbb{R}$ be a function, with epigraph $\operatorname{epi}f\subseteq\mathbb{R}^{n+1}$.

(i) $f$ is convex if and only if $\operatorname{epi}f$ is a convex subset of $\mathbb{R}^{n+1}$.