Transfer continuities, generalizations of the Weierstrass and maximum theorems: A full characterization
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This paper gives necessary and sufficient conditions for 1. (1) a function to attain its maximum on a compact set 2. (2) the set of maximum points of a function on a compact set to be non-empty and compact, and 3. (3) the maximum (marginal) correspondence to be closed. We do so by introducing a class of transfer continuities which characterize the essence of topological structures of functions and correspondences for extreme points and significantly weaken the conventional continuities. Thus our results generalize the classical Weierstrass theorem and the Maximum Theorem of Berge (Espaces topologiques et fonctions multivoques, Donod, Paris, 1959; Topological Spaces, Macmillan, New York, 1963, p. 116), by giving necessary and sufficient conditions. Furthermore, we generalize the maximum theorem of Walker, (International Economic Review, 1979, 20, 267-270) by relaxing the openness of the graph of preference correspondences and the lower semi-continuity of the feasible action correspondence. By applying our maximum theorems to game theory and economics, we can generalize many of the existence theorems on Nash equilibrium of games and equilibrium of the generalized games (the so-called abstract economies) in the literature. 1995.