1.2 Epigraphs

Epigraph

The epigraph of a function $f:X\to \bar{\mathbb{R}}$ is the set of points on or above $f$. Formally,
$\mathrm{epi}f=\{(x,t)\in X\times \mathbb{R}:f(x)\le t\}.$
Similarly, the interior of $f$ can be defined as the strict (open) epigraph (i.e., the points in the range of $f$ are excluded from the set).