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.

References

[a1]	R.T. Rockafellar, "Convex analysis" , Princeton Univ. Press (1970) pp. 23; 307