GENERALIZING GENERIC DIFFERENTIABILITY PROPERTIES FROM CONVEX TO LOCALLY LIPSCHITZ FUNCTIONS

David Preiss proved that every locally Lipschitz function on an open s ubset of a Banach space which has an equivalent norm Gateaux (Frechet) differentiable away from the origin is Gateaux (Frechet) differentiab le on a dense subset of its domain. It is known that every continuous convex function on an open convex subset of such a space is Gateaux (F rechet) differentiable on a residual subset of its domain. We show tha t for a locally Lipschitz function on a separable Banach space (with s eparable dual) there are residual subsets which if the function were c onvex would coincide with its set of points of differentiability. Thes e are the sets where the function is fully intermediately differentiab le (fully and uniformly intermediately differentiable) and sets where the subdifferential mapping is weak (norm) lower semi-continuous. We discuss the role of these sets in generating the subdifferential and p resent a refinement of Preiss result. (C) 1994 Academic Press, Inc.