# Smooth and proper map question [on hold]

Let $f: U\to \mathbb{R}^n$ be smooth and proper map such that det$df_x$ does not change sign outside some compact subset of $U$. Prove that $f$ is one-to-one and $\exists$ $x \in U$ such that $f(x) = 0$.

Some help please =)

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## put on hold as off-topic by Michael Albanese, Daniel Rust, Avitus, Micah, Bungoyesterday

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Michael Albanese, Daniel Rust, Avitus, Micah, Bungo
If this question can be reworded to fit the rules in the help center, please edit the question.

If you browse a bit the questions and answers on this site, you'll immediately notice that people do not write "plz" and such. This is not SMS, in fact! –  Mariano Suárez-Alvarez Apr 1 '11 at 1:39
What is $U$? Presumably it's $n$-dimensional, so that $df_x$ even might have a determinant in the first place? In any case, it doesn't seem right to me that 0 would have to be in the image of $f$, unless $f$ ends up in fact having to be surjective... –  Aaron Mazel-Gee Apr 1 '11 at 8:15
oops... you´re wright. –  Júlio César Apr 1 '11 at 16:36
This is not an answer but just a comment, I would have wrote this as a comment if I had had enough reputation. 1)You should restrict yourself to a connected and n-dimensional manifold $U$, otherwise your thesis has not sense or is obviously false. 2)For easiness start assuming that $f$ is everywhere regular. 3)Because the hypothesis are preserved adding to $f$ any constant function, your thesis really means that $f$ should be bijective. In these terms your problem sounds: for a local diffeomorphism $f:U\to\mathbb{R}^n$, where $U$ is a n-dimensional connected manifold, does properness i –  Giuseppe Tortorella Apr 1 '11 at 20:36
The comment of Giu was converted from an answer, but the end was cut off at the beginning of the word "imply". It ends with: "does properness imply bijectivity?". –  Jonas Meyer Apr 4 '11 at 18:28