A chain rule for essentially smooth Lipschitz functions

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 Rⁿ, then g ○ f is strictly differentiable almost everywhere in Rⁿ, where f ≡ (f₁,f₂,...,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.