Differentiability of quasiconvex functions on separable Banach spaces

We investigate the differentiability properties of a real-valued quasiconvex function 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 GA cent teaux 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 GA cent teaux 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 a"e (N) is Fr,chet differentiable a.e.