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 Frechet 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}(a) 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 (classical) sense of Kempisty since this interesting property does not seem to have been noticed before.

💡 Research Summary

The paper investigates the continuity properties of real‑valued quasiconvex functions defined on Baire topological vector spaces, a class that includes Banach and Fréchet spaces. The central question is: under what purely topological conditions does a quasiconvex function (f) become continuous on a residual (i.e., comeager) subset of the space? The authors answer this by introducing a simple yet powerful criterion that involves only the lower level sets of (f).

A quasiconvex function is one for which every sub‑level set
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