# Supergraph

of a function

Let $f: X \rightarrow \overline{\mathbf R}\;$ be a function defined on some set $X$ with values in the extended real line $\overline{\mathbf R}\; = \mathbf R \cup \{ - \infty \} \cup \{ + \infty \}$. Then the supergraph of $f$ is the subset of the Cartesian product $X \times \mathbf R$ consisting of all points $( x, \alpha )$ lying "above" the graph of $f$. The supergraph is sometimes called the epigraph and is denoted by $\mathop{\rm epi} f$:

$$\mathop{\rm epi} f = \ \{ {( x, \alpha ) \in X \times \mathbf R } : {\alpha \geq f ( x) } \} .$$

The projection of the supergraph onto $X$ is known as the effective domain of $f$, denoted by $\mathop{\rm dom} f$:

$$\mathop{\rm dom} f = \{ {x \in X } : {f ( x) < + \infty } \} .$$

The function $f$ is said to be proper if

$$f ( x) > - \infty , \forall x,\ \textrm{ and } \ \ \mathop{\rm dom} f \neq \emptyset .$$

A function $f: X \rightarrow \mathbf R$ defined on a real vector space $X$ is convex if and only if $\mathop{\rm epi} f$ is a convex subset of $X \times \mathbf R$. A function $f: X \rightarrow \overline{\mathbf R}\;$ defined on a topological space $X$ is upper semi-continuous if and only if $\mathop{\rm epi} f$ is a closed set.