Continuity properties of the Nash equilibrium correspondence in a discontinuous setting

We introduce a new complete metric space of discontinuous normal form games and prove that the Nash equilibrium correspondence is upper semicontinuous with non-empty and compact values. So, using the Theorem of Fort (1949), we obtain that the correspondence is also lower semicontinuous in a dense subset. We introduce new topological assumptions on the payoff functions and a strengthening of standard quasi-concavity properties. Examples show that our results cannot be obtained from the previous ones.