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.

Consider a game with ${\displaystyle N}$ agents with agent ${\displaystyle i}$ having strategy ${\displaystyle A_{i}\subseteq {\mathbb {R}}}$; write ${\displaystyle \mathbf {a} }$ for an N-tuple of actions (i.e. ${\displaystyle \mathbf {a} \in \prod _{j=1}^{N}A_{j}}$) and ${\displaystyle \mathbf {a} _{-i}=(a_{1},a_{2},\ldots ,a_{i-1},a_{i+1},\ldots ,a_{N})}$ as the vector of all agents' actions apart from agent ${\displaystyle i}$.

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