Is this a local diffeomorphism?

Question

I want to find a local diffeomorphism $\Bbb{R}^2\to\Bbb{R}^2$ that is not a diffeomorphism onto its image. This is what I thought: $f(x,y)=(\sin 2\pi x, \cos 2\pi y)$. Does that work? Seems ok to me.

Answer 1

$\left(x,y\right)\mapsto\left(e^{x}sin\left(y\right),e^{x}cos\left(y\right)\right)$