On stepanov Type differentiability Theorems

Abstract

The main result shows that the Rademacher theorem proved by J. Lindenstrauss and D. Preiss [13] (which says that, for some pairs X, Y of Banach spaces, each Lipschitz f : X → Y is $${\Gamma}$$ -a.e. Frechet differentiable) generalizes the corresponding stepanov theorem (whichsays that, forsuch X and Y, an arbitrary f : X → Y is Frechet differentiable at $${\Gamma}$$ -almost all points at which f is Lipschitz). We also present an abstract approach which shows an easy way how (in some cases) a theorem of stepanov type (for vector functions) can be inferred from the corresponding theorem of Radamacher type. Finally we present Stepanov type differentiability theorems with the assumption of pointwise directional Lipschitzness.