Existence of Nash equilibria for generalized games without upper semicontinuity

The present note extends Debreu's equilibrium existence theorem for a generalized game in the context of finite-dimensional strategy spaces, by weakening the upper semicontinuity and closed-valuedness assumption on the feasible strategy multifunctions. This is made by establishing an inequality of Ky Fan's type, whose proof is based on a selection theorem by E. Michael. An extension to generalized games with unbounded strategy spaces is also presented.