A Chain Rule for Essentially Smooth Lipschitz Functions

Abstract

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 : Rm \rightarrow R$ is arcwise essentially smooth on Rm and each function $f_j : R^n \rightarrow R, 1 \leq j \leq m$, is strictly differentiable almost everywhere in Rn, then $g \circ f$ is strictly differentiable almost everywhere in Rn, where $f \equiv (f_1,f_2,\ldots,f_m)$. 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.