Regular nonsmooth equations (Q1322711)

Notions of regular points and values for continuous (and not necessarily smooth) functions introduced by \textit{T. Rader} [Econometrica 41, 913-935 (1973; Zbl 0278.90007)] are used. It is shown that if \(y\) is a regular value of a continuous function \(f\), then the solutions to the equation \(f(x) = y\) are locally unique, and the degree \(d(f,A,y)\) of \(f\) on an open set \(A\) at \(y\) is given by \[ d(f,A,y) = \sum_{x \in f^{-1} (y)} \text{sgn det } Df(x). \] These results are used to extend \textit{E. Dierker's} results [Econometrica 40, 951-953 (1972; Zbl 0259.90005)] on the number of equilibria in regular exchange economies to nonsmooth exchange economies.