This paper generalizes the Theorem of the Maximum (Berge, 1963) to allow for discontinuous changes in the domain and the objective function. It also provides a geometrical version of the (generalized) theorem.

) and Shaked and Sutton (1982). Finally, despite the fact that the equilibrium concept is open-loop, all but the introductory price of the high quality good converge to marginal cost in the limit as firms can change prices arbitrarily frequently.