Points of Continuity of Quasiconvex Functions on Topological Vector Spaces

We give necessary and sufficient conditions for a real-valued quasiconvex function f on a Baire topological vector space X (in particular, Banach or Fréchet space) to be continuous at the points of a residual subset of X. These conditions involve only simple topological properties of the lower level sets of f. A main ingredient consists in taking advantage of a remarkable property of quasiconvex functions relative to a topological variant of essential extrema on the open subsets of X. One application is that if f is quasiconvex and continuous at the points of a residual subset of X, then with a single possible exception,

is nowhere dense or has nonempty interior, as is the case for everywhere continuous functions. As a barely off-key complement, we also prove that every usc quasiconvex function is quasicontinuous in the (topological) sense of Kempisty since this interesting property does not seem to have been noticed before.