Weak univalence and connectedness of inverse images of continuous functions

A continuous function f with domain X and range f(X) in R-n is weakly univa lent if there is a sequence of continuous one-to-one functions on X converg ing tof uniformly on bounded subsets of X. In this article, we establish, u nder certain conditions, the connectedness of an inverse image f(-1)(q). Th e univalence results of Radulescu-Radulescu, More-Rheinboldt, and Gale-Nika ido follow from our main result. We also show that the solution set of a no nlinear complementarity problem corresponding to a continuous P-0-function is connected if it contains a nonempty bounded clopen set; in particular, t he problem will have a unique solution if it has a locally unique solution.