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Hello. I have the next problem:

Let $f: U\subset R^n\rightarrow R$ a class $C^1$ function in an open subset $U$ of $R^n$. Proff that f can't be injective.

There are some idications: suppose that the vector ($\nabla f$)(p) is not zero (if it's zero the function is not injective WHY?) in $p\in U$ and that we can assume that $\frac{\partial f}{\partial x_i}$(p) $\neq$ 0; consider so $F(x_1,x_2,...,x_n) = (f(x),x_2,...x_n)$ with $x=(x_1,x_2,..,x_n)$ and get the result applying the inverse function theorem.

Thank you so much for any help!

Let $f: U\subset R^n\rightarrow R$ a class $C^1$ function in an open subset $U$ of $R^n$. Proff that f can't be injective.

There are some idications: suppose that the vector ($\nabla f$)(p) is not zero (if it's zero the function is not injective WHY?) in $p\in U$ and that we can assume that $\frac{\partial f}{\partial x_i}$(p) $\neq$ 0; consider so $F(x_1,x_2,...,x_n) = (f(x),x_2,...x_n)$ with $x=(x_1,x_2,..,x_n)$ and get the result applying the inverse function theorem.

Thank you so much for any help!

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