On Singularities of Mappings of Euclidean Spaces. I. Mappings of the Plane Into the Plane

Abstract

Let f 0 be a mapping of an open set R in n-space E n into m-space E m . Let us consider, along with f 0, all mappings f which are sufficiently good approximations to f 0. By the Weierstrass Approximation Theorem, there are such mappings f which are analytic; in fact, (see [5], Lemma 6) we may make f approximate to f 0 throughout R arbitrarily well, and if f 0 is r-smooth (i.e., has continuous partial derivatives of orders ≦r), r finite, we may make corresponding derivatives of fapproximate to those of f 0.