Journal of Convex Analysis 20 (2013), No. 3, 701--721
Copyright Heldermann Verlag 2013
Points of Continuity of Quasiconvex Functions on Topological Vector Spaces
Patrick J. Rabier
Dept. of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A.
rabier@imap.pitt.edu
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.
Keywords: Quasiconvex function, topological essential extremum, point of continuity, point of discontinuity, Baire category, quasicontinuity.
MSC: 26B05, 52A41, 54E52
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