# Supergraph

From Encyclopedia of Mathematics

*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.

#### 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=17155

This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article