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of a function

Let be a function defined on some set with values in the extended real line . Then the supergraph of is the subset of the Cartesian product consisting of all points lying "above" the graph of . The supergraph is sometimes called the epigraph and is denoted by :

The projection of the supergraph onto is known as the effective domain of , denoted by :

The function is said to be proper if

A function defined on a real vector space is convex if and only if is a convex subset of . A function defined on a topological space is upper semi-continuous if and only if is a closed set.



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