What is an example of a smooth function with continuous derivatives, that is locally invertible but not globally, and the reason for that is not injectivity.
My first idea was $f:\mathbb{R}^{2}\to \mathbb{R}^{2}$ defined by $f(x,y)=(e^{x}\cos y, e^{x}\sin y)$ everything above is satisfied except that the function is injective...
What example one can take?
Thanks
$f$ is not injective because $f(x,y) = f(x,y+2\pi)$. Your example is correct.