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With utmost pleasure, I dedicate this article to my teacher, mentor, and friend Olvi Mangasarian on the occasion of his 70th birthdayIn this article, we prove inverse and implicit function theorems for H -differentiable functions, thereby giving a unified treatment of such theorems for C 1 -functions, PC 1 -functions, and for locally Lipschitzian functions. We also derive inverse and implicit function theorems for semismooth functions.

In two recent papers, Facchinei 7] and Facchinei and Kanzow 8] have shown that for a continuously di erentiable P 0-function f , the nonlinear complementarity problem NCP(f ") corresponding to the regularization f " (x) := f (x) + "x has a unique solution for every " > 0, that dist (x("); SOL(f)) ! 0 as " ! 0 when the solution set SOL(f) of NCP(f) is nonempty and bounded, and NCP(f) is stable if and only if the solution set is nonempty and bounded. They prove these results via the the Fischer function and the Mountain Pass Theorem. In this paper, we generalize these NCP results to a Box Variational Inequality Problem corresponding to a continuous P 0-function where the regularization is described by an integral. We also describe an upper semicontinuity property of the inverse of a weakly univalent function and study its consequences.