Definition:Epigraph

Definition

Let $f:S\to\R \cup \left\{{+\infty, -\infty}\right\}$ be an extended-real-valued function.

The epigraph of $f$ is the set:

$\operatorname{epi}f := \left\{ \left(x,\alpha\right)\in S \times \R,\ f(x)\leq \alpha \right\}$