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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.

Comments

References

[a1] R.T. Rockafellar, "Convex analysis" , Princeton Univ. Press (1970) pp. 23; 307
How to Cite This Entry:
Supergraph. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Supergraph&oldid=48912
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article