  We investigate the differentiability properties of real-valued quasiconvex
functions f defined on a separable Banach space X. Continuity is only assumed
to hold at the points of a dense subset. If so, this subset is automatically
residual. Sample results that can be quoted without involving any new concept
or nomenclature are as follows: (i) If f is usc or strictly quasiconvex, then f
is Hadamard differentiable at the points of a dense subset of X (ii) If f is
even, then f is continuous and Gateaux differentiable at the points of a dense
subset of X. In (i) or (ii), the dense subset need not be residual but, if X is
also reflexive, it contains the complement of a Haar null set. Furthermore,
(ii) remains true without the evenness requirement if the definition of Gateaux
differentiability is generalized in an unusual, but ultimately natural, way.
The full results are much more general and substantially stronger. In
particular, they incorporate the well known theorem of Crouzeix, to the effect
that every real-valued quasiconvex function on R^N is Frechet differentiable
a.e.
