Asplund sets, differentiability and subdifferentiability of functions in Banach spaces ✩

We show that Asplund sets are effective tools to study differentiability of Lipschitz functions, and ε-subdifferentiability of lower semicontinuous functions on general Banach spaces. If a locally Lipschitz function defined on an Asplund generated space X = T Y has a minimal Clarke subdifferential mapping, then it is T BY -uniformly strictly differentiable on a dense Gδ subset of X. Examples are given of locally Lipschitz functions that are T BY -uniformly strictly differentiable everywhere, but nowhere Fréchet differentiable. © 2005 Elsevier Inc. All rights reserved.