Can I draw this conclusion about invertibility of a function?

Question

Suppose we have a smooth homeomorphism $f:\mathbb{R}^m\rightarrow A\subseteq\mathbb{R}^n$ such that $Df_p\neq 0$ for any $p$. Can we conclude that $f^{-1}$ is smooth, in these conditions? I think so, because we can apply the Inverse Function Theorem and get a smooth inverse locally in each point of $U$. As smoothness is a local definition, we can conclude $f^{-1}$ is globally smooth. Is this true?

Answer 1

Define $f : \mathbb{R}^2 \to \mathbb{R}^2$ by $f(x,y) = (x,y^3)$. This is a smooth map, and a homeomorphism. Observe the first partial $\frac{\partial f}{\partial x}$ is identically $1$, and so the Jacobian $Df$ never vanishes. However, this is not a diffeomorphism.