In this paper we address some of the most fundamental questions regarding the differentiability structure of locally Lipschitz functions defined on separable Banach spaces. For example, we examine the relationship between integrability, D-representability, and strict differentiability. In addition to this, we show that on any separable Banach space there is a significant family of locally Lipschitz functions that are integrable, D-representable and possess desirable differentiability properties. We also present some striking applications of our results to distance functions. Academic Press

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We study a class of Banach spaces which have the property that every continuous convex function on an open convex subset of the dual possessing a weak * continuous subgradient at points of a dense G § subset of its domain, is Frechet differentiate on a dense G$ subset of its domain. A smaller more amenable class consists of Banach spaces where every minimal weak * cusco from a complete metric space into subsets of the second dual which intersect the embedding from a residual subset of the domain is single-valued and norm upper semi-continuous at the points of a residual subset of the domain. It is known that all Banach spaces with the Radon-Nikodym property belong to these classes as do all with equivalent locally uniformly rotund norm. We show that all with an equivalent weakly locally uniformly rotund norm belong to these classes. The condition closest to a characterisation is that the Banach space have its weak topology fragmentable by a metric whose topology on bounded sets is stronger than the weak topology. We show that the space ^oo(Γ), where Γ is uncountable, does not belong to our special classes.

In this paper we introduce a new class of real-valued locally Lipschitz functions (that are similar in nature and definition to Valadier's saine functions), which we call arcwise essentially smooth, and we show that if g : R m → R is arcwise essentially smooth on R m and each function f j : R n → R, 1 ≤ j ≤ m, is strictly differentiable almost everywhere in R n , then g • f is strictly differentiable almost everywhere in R n , where f ≡ (f 1 , f 2 ,. .. , fm). We also show that all the semismooth and all the pseudoregular functions are arcwise essentially smooth. Thus, we provide a large and robust lattice algebra of Lipschitz functions whose generalized derivatives are well behaved.

In this paper we extend the notion of a Lebesgue-null set to a notion which is valid in any completely metrizable Abelian topological group. We then use this definition to introduce and study the class of essentially smooth functions. These are, roughly speaking, those Lipschitz functions which are smooth (in each direction) almost everywhere.