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Weak univalence and connectedness of inverse images of continuous functions. (English) Zbl 0977.90060
Summary: A continuous function $$f$$ with domain $$X$$ and range $$f(X)$$ in $$R^n$$ 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 $$P_0$$-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.

##### MSC:
 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
##### Keywords:
weak univalence; connectedness; complementarity problem
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