  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.
