Abstract
Recently, Borwein and Moors introduced a new class of real-valued locally Lipschitz functions, that are similar in nature and definition to Valadier's saine functions, which they called arc-wise essentially smooth. They showed that if g n M is an arc-wise essentially smooth real-valued function and f m M n is strictly differentiable almost everywhere, then g f m M is also strictly differentiable almost everywhere. They also showed that this class possesses strong closure properties. In this paper, we give an appropriate extension of this class to locally Lipschitz mappings defined between Banach spaces. We show that the results established by Borwein and Moors in the finite dimensional setting also hold for arc-wise essentially smooth mappings defined between Banach spaces.