On Sun, 28 Nov 2010 20:30:03 +0000 (UTC), Rodolfo Conde <email@example.com> wrote:
> > > Hi all, > > I am trying to find a counterexample for the following >proposition (which is false): > > Suppose that we have a sequence (f_n: K -> R^q) (where K is a >compact subset of some euclidean space R^n) of continuous functions and this >sequence converges uniformly to a function f: K -> R^q such that f is an >imbedding of K. Then for n large enough, f_n is an imbedding
I assume by imbedding you mean at least that f is a homeomorphism to ite image. Here is a straightforward counterexample in one dimension: Let K be the interval [0,1] and define f_n : K -> R by f_n(x) = 0, 0 <= x < 1/n; f_n(x) = x - 1/n, 1/n <= x <= 1. This converges uniformly to the identity function f(x) = x.
One could clearly adapt this to provide an example for any dimension. And one could adapt it to give f_n any number of derivatives.
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