
Journal of Convex Analysis 13 (2006), No. 3, 695709 Copyright Heldermann Verlag 2006 εFréchet Differentiability of Lipschitz Functions and Applications Marián Fabian Mathematical Institute, Academy of Sciences, Zitná 25, 115 67 Prague 1, Czech Republic fabian@math.cas.cz Philip D. Loewen Dept. of Mathematics, University of British Columbia, Vancouver, B. C., Canada V6T 1Z2 loew@math.ubc.ca Xianfu Wang Dept. of Mathematics & Statistics, Univ. of B. C. at Okanagan, 3333 University Way, Kelowna, B. C., Canada V1V 1V7 Shawn.Wang@ubc.ca We study the εFréchet 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 εFréchet 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 εFréchet differentiability is separably reducible. Keywords: epsilonFrechet differentiability, meanvalue theorem, local epsilonsupport, intermediate differentiability, Asplund generated space, boundedly Asplund set, separable reduction. MSC: 46G05; 58C20, 49J50 [ Fulltextpdf (449 KB)] for subscribers only. 