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 ( y α ) α of games converging to y and a net ( x α ) α such that x α is a Nash equilibrium of any y α , then at least a cluster point of ( x α ) 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.