Do I misunderstand the definition of diffeomorphism?

Question

I am reading Differential geometry of curves and surfaces by Do Carmo.But I am confused of the difinition of diffeomorphism: A differential mapping $F:V⊂R^{n}→W⊂R^{n}$, where $V$ and $W$ are open sets, is called a diffeomorphism of $V$ with $W$ if $F$ has a differentiable inverse.

I am confused by the differientiable inverse. Does that mean you can define $F^{-1}:W→V$ where all points in $W$ are differentiable, or does it mean you only can find some subsets of $W$ that are differentiable?

Answer 1

It means two things:

1. The function $F\colon\, V\longrightarrow W$ is a differentiable function which has an inverse $F^{-1}\colon\, W\longrightarrow V$.
2. The function $F^{-1}\colon W\longrightarrow V$ is differentiable, too.

Answer 2

$F^{-1}$ has to be differentiable evrywhere. If it's differentiable only on a subset $\tilde W \subset W$, you may be able to find $\tilde V\subset V$ such that $F(\tilde V) \subset \tilde W$. Then $F|_{\tilde V}: \tilde V \rightarrow F(\tilde V)$ may be a diffeomorphism (if it satisfies the rest of the conditions), but $F$ is not.