Fréchet differentiability of regular locally Lipschitzian functions

AbstractThis paper considers Fréchet differentiability almost everywhere in the sense of category of regular, locally Lipschitzian real-valued functions defined on open subsets of a Banach space. It is first shown that, for separable Banach spaces, Clarke's generalized gradient of such a function is a minimal, convex- and compact-valued, upper semicontinuous multifunction. Using a theorem of Christensen and Kenderov it is then shown that, for separable Asplund spaces, such a function is Fréchet differentiable on a dense Gδ subset of its domain