An implicit-function theorem for a class of nonsmooth functions

This paper introduces the concept of strong approximation of functions, and a related concept of strong Bouligand (B-) derivative, and these ideas are employed to prove an implicit-function theorem for nonsmooth functions. This theorem provides the same kinds of information as does the classical implicit-function theorem, but with the classical hypothesis of strong Fréchet differentiability of the implicit function. Therefore it is applicable to a considerably wider class of functions than is the classical theorem. In the last part of the paper this implicit function result is applied to analyze local solvability and stability of perturbed generalized equations.