We give necessary and sufficient conditions for a real-valued quasiconvexfunction f on a Baire topological vector space X (in particular, Banach orFrechet space) to be continuous at the points of a residual subset of X. Theseconditions involve only simple topological properties of the lower level setsof f. A main ingredient consists in taking advantage of a remarkable propertyof quasiconvex functions relative to a topological variant of essential extremaon the open subsets of X. One application is that if f is quasiconvex andcontinuous at the points of a residual subset of X, then with a single possibleexception, f^{-1}(a) is nowhere dense or has nonempty interior, as is the casefor everywhere continuous functions. As a barely off-key complement, we alsoprove that every usc quasiconvex function is quasicontinuous in the (classical)sense of Kempisty since this interesting property does not seem to have beennoticed before.
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