# A chain rule for essentially strictly differentiable Lipschitz functions

Borwein, Jonathan M. and Moors, Warren B. (1998) A chain rule for essentially strictly differentiable Lipschitz functions. SIAM Journal on Optimization, 8 (2). pp. 300-308.

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In this paper we introduce a new class of real-valued locally Lipschitz functions, (that are similar in nature and definition to Valadier's {\sl saine} functions) which we call {\sl arc-wise essentially smooth}, and we show that if $g : R^n \rightarrow R$ is arc-wise essentially smooth on $R^n$ and each function $f_j : R^m \rightarrow R,\ 1 \leq j \leq n$ is strictly differentiable almost everywhere in $R^m$, then $g \circ f$ is strictly differentiable almost everywhere in $R^m$, where $f \equiv (f_1,f_2, . . . f_n)$. We also show that all the semi-smooth and pseudo-regular functions are arc-wise essentially smooth. Thus, we provide a large and robust lattice algebra of Lipschitz functions whose generalized derivatives are well-behaved.