Points of Continuity of Quasiconvex Functions on Topological Vector Spaces

P. J. Rabier

Journal of convex analysis, Tome 20 (2013) no. 3, pp. 701-721.

Résumé

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, f^-1(α) 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.

Détail

Classification : 26B05, 52A41, 54E52
Mots-clés : Quasiconvex function, topological essential extremum, point of continuity, point of discontinuity, Baire category, quasicontinuity