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Apply Implicit Function Theorem to show that no $C^1$ function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ can be one to one near any point of its domain. Repeat the proof by using Inverse Mapping Theorem instead of IFT.

Assume $\frac{df}{dx}(a,b)\neq 0$ and define $g(x,y)=(f(x,y),y)$. Prove that $g$ is a one-to-one function near the point $(a,b)$.

I know I am supposed to give some of my work first, but I have spent hours on reading the book about IFT and I have no idea at all about this question :( can anyone at least give me some hints or explain the question ?

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Think of the level sets $f(x,y)=c$. If you look very closely, like with a microscope, the level sets will be parallel lines. Using the implicit function theorem, you can find a non-linear coordinate transformation of the xy-plane so that also in the macroscopic picture the level sets are parallel lines. Well, parallel lines of the same function value are a direct contradiction to bijectivity.

The only exception may happen where the level sets form concentric circles around a singular point of $f$. But even then you do not have bijectivity in the neighborhood of the singular point.

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