Let $f:U\subset\mathbb{R}^m\rightarrow\mathbb{R}^n$ be a function of class $C^1$ with $m> n$. Prove that $f$ is not an injection function.
I do not know how to solve this problem. Could someone help me with the solution to this problem?
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1Idk the answer but I guess that it (probably) is related to the property that differentiability is a local property and that the derivative have a linear local approximation... but as I says it is just a guess (probably wrong). – Masacroso Aug 01 '17 at 01:11
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I thought first of doing it as absurd, assuming f is injective and then trying to relate in some way to the local form of the submersions, but I do not know how to do it, since I do not know if it is an isomorphism. – Matematico Aug 01 '17 at 01:14
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This is referred to as Topological invariance of dimension see the post by Terry Tao, Corollary 3 – Weaam Aug 01 '17 at 01:19
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Interesting. It's the same problem. – Matematico Aug 01 '17 at 01:22
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The blog post doesn't assume continuous differentiability though, so there's probably an easier proof under a $C^1$ assumption. – Weaam Aug 01 '17 at 01:23
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Truth. I did not realize that no post did not ask to be C1 – Matematico Aug 01 '17 at 01:25
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Here you go https://math.stackexchange.com/questions/1086314/continuously-differentiable-map-from-mathbbrmn-to-mathbbrn/1086462#1086462 – Reveillark Aug 01 '17 at 01:29