A CHAIN RULE FOR ESSENTIALLY SMOOTH LIPSCHITZ FUNCTIONS

In this paper we introduce a new class of real-valued locally Lipschit z functions (that are similar in nature and definition to Valadier's s aine 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 f unction f(j) : R-n --> R, 1 less than or equal to j less than or equal to m, is strictly differentiable almost everywhere in R-n, then g cir cle f is strictly differentiable almost everywhere in R-n, where f = ( f(1), f(2), ..., f(m)). We also show that all the semismooth and all t he pseudoregular functions are arcwise essentially smooth. Thus, we pr ovide a large and robust lattice algebra of Lipschitz functions whose generalized derivatives are well behaved.