Epigraph

Remarks

Only real numbers $r$ are used, so points $x$ with $f(x) = +\infty$ contribute nothing, and points with $f(x) = -\infty$ contribute the whole line $\set{x} \times \R$. The strict epigraph uses $f(x) < r$ instead. If $X$ is a real vector space, $f$ is convex if and only if $\operatorname{epi} f$ is a convex set. If $X$ is a topological space, $f$ is lower semicontinuous if and only if $\operatorname{epi} f$ is closed in $X \times \R$. The function is recovered as $f(x) = \inf \set{r \in \R : (x, r) \in \operatorname{epi} f}$, with $\inf \emptyset = +\infty$.

Example: for $f(x) = x^2$ on $\R$, the point $(1, 2)$ lies in $\operatorname{epi} f$, since $f(1) = 1 \le 2$.

Non-example: the point $(2, 1)$ does not lie in $\operatorname{epi} f$, since $f(2) = 4 > 1$.