Hi guys... Haven't been in the forum for a couple years now.(adsbygoogle = window.adsbygoogle || []).push({});

I have an old analysis problem that I never manage to solve. Would be nice if someone can shed some light on this.

let [tex]f[/tex] be a [tex]C^1[/tex] function from [tex]\mathbb{R}^n \rightarrow \mathbb{R}^n[/tex], [tex]n>1[/tex]. [tex]df[/tex] is invertible except at isolated points (WLOG assume only at 0), prove that [tex]f[/tex] is locally injective, i.e. there is a neighborhood around each point in the domain such that [tex]f[/tex] is injective.

thoughts about this problem: contraction principle doesn't work at all, since df gets really small around 0. The theorem is false when n=1 (like y=x^2), this makes me think the problem should involve some (if not mostly) topology of R^n

I asked my topology professor but he said he can't think of a solution right away, he told me to consider the eigenvectors of the derivative... would be nice if anyone can put this problem to "sleep" once and for all.

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# Multivariable calculus/analysis problem

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