Graph continuous function
In mathematics, and in particular the study of game theory, a function is graph continuous if it exhibits the following properties. The concept was originally defined by Partha Dasgupta and Eric Maskin in 1986 and is a version of continuity that finds application in the study of continuous games. Notation and preliminaries
Consider a game with  agents with agent  having strategy ; write  for an N-tuple of actions (i.e. ) and  as the vector of all agents' actions apart from agent . Let  be the payoff function for agent . A game is defined as . Definition

Function  is graph continuous if for all  there exists a function  such that  is continuous at . Dasgupta and Maskin named this property "graph continuity" because, if one plots a graph of a player's payoff as a function of his own strategy (keeping the other players' strategies fixed), then a graph-continuous payoff function will result in this graph changing continuously as one varies the strategies of the other players. The property is interesting in view of the following theorem. If, for ,  is non-empty, convex, and compact; and if  is quasi-concave in , upper semi-continuous in , and graph continuous, then the game  possesses a pure strategy Nash equilibrium. References

 Partha Dasgupta and Eric Maskin 1986. "The existence of equilibrium in discontinuous economic games, I: theory". The Review of Economic Studies, 53(1):1–26

Game theory
Theory of continuous functions