The following question was part of my quiz on smooth manifolds and I couldn't solve it. I tried again at home but in vain.
A smooth map $f: M\to N$ between manifolds is said to be a local diffeomorphism if, around each point $p \in M$ there exists a nbd $U_p$ of p such that $U_p$ is diffeomorphic to its image under f.
(a) Prove that any local diffeomorphism $f: \mathbb{R} \to \mathbb{R}$ is diffeomorphism onto its image.
(b) Give an example to show that this fact is not true if we consider a smooth map from $\mathbb{R}^2 $ to $\mathbb{R}^2$.
So, it is given that for every $p\in M$ there exists a nbd $U_p$ of p such that $U_p$ is diffe. to its image which implies: f is $C^1$ , $f(U_p)$ is open, $f^{-1} $ is $C^1$ but there might be for point p' a nbd $U_p'$ where a different diffeomorphism f' exists. But I have to prove the existence of a single function (Say F) and I am not getting any intuition.
So, can you please help?