Weak Univalence and Connectedness of Inverse Images of Continuous Functions

A continuous function f with domain X and range f(X) in Rⁿ is weakly univalent if there is a sequence of continuous one-to-one functions on X converging to f uniformly on bounded subsets of X. In this article, we establish, under certain conditions, the connectedness of an inverse image f⁻¹(q). The univalence results of Radulescu-Radulescu, Moré-Rheinboldt, and Gale-Nikaido follow from our main result. We also show that the solution set of a nonlinear complementarity problem corresponding to a continuous P₀-function is connected if it contains a nonempty bounded clopen set; in particular, the problem will have a unique solution if it has a locally unique solution.