ε-Fréchet Differentiability of Lipschitz Functions and Applications

We study the ε-Fréchet differentiability of Lipschitz functions on Asplund generated Banach spaces. We prove a mean valued theorem and its equivalent, a formula for Clarke's subdifferential, in terms of this concept. We inspect proofs of several statements based on the deep Preiss's theorem on Fréchet differentiability of Lipschitz functions and we recognize that it is enough to use a simpler lemma on ε-Fréchet differentiability due to Fabian and Preiss. We do so for generic differentiability results of Giles and Sciffer, for the existence of nearest points of Borwein and Fitzpatrick, etc. We also show that the ε-Fréchet differentiability is separably reducible.