© 1996, Topology Atlas

Let U and V be topological (or convergence) spaces and f be a function from U x V to the extended real line. Whenever one fixes a point u in U, the infimum (resp. the supremum) of f (u, ) in V can be considered, so two functions can be constructed: the function g which associates to any point u of U the infimum of f (u, ) in V and the function h which associates to any point u of U the supremum of f (u, ) in V. The function g and the function h are called marginal functions. The first results on this topic are, at my knowledge, due to C. Berge and can be founded in his book: Topological spaces, Mac Millan, New York (1963). The marginal functions have been studied from several points of view: differentiability, semicontinuity, convergence, convexity, etc. A branch of topology, namely the theory of multifunctions, plays a fundamental role in investigating marginal functions arising from constrained parametric optimization problems. During last years, these functions have been used mostly in Multilevel Optimization, in Nonsmooth Analysis and in Game Theory.