Let $M$ and $N$ be surfaces, and $f: M\rightarrow N$ a differentiable and bijective map
(a) Can you ensure that f is a diffeomorphism?
(b) Suppose further that the map $T_{p}f:T_{M}\rightarrow T_{f(p)}N$ is an isomorphism for each $p\in M$, can you ensure that a diffeomorphism?
in (a) i think that is false, but I try to make a example, but the differentiability of the inverse is always hard