Hadamard well-posedness in discontinuous non-cooperative games

We consider some recent classes of discontinuous games with Nash equilibria and we prove that such classes have the Hadamard well-posedness property. This means that given a game y, a net (yα)α of games converging to y and a net (xα)α such that xα is a Nash equilibrium of any yα, then at least a cluster point of (xα)α is a Nash equilibrium of y. In order to obtain this property, we prove that the map of Nash equilibria is upper semicontinuous. Using the pseudocontinuity, a generalization of the continuity, we improve previous results obtained with continuous functions.