  In this paper we consider continuity of the set of Nash equilibria and
approximate Nash equilibria for parameterized games. For parameterized games
with unique Nash equilibria, the continuity of this equilibrium mapping is
well-known. However, when the equilibria need not be unique, there may exist
discontinuities in the equilibrium mapping. The focus of this work is to
summarize continuity properties for parameterized Nash equilibria and prove
continuity via the approximate Nash game with uniformly continuous objective
functions over potentially non-compact strategy spaces.
