Let $M$ be a topological manifold and let $(U,\phi)$ and $(V,\psi)$ be two charts. This post points to condition $(1)$ below implying condition $(2)$:
- $\phi\circ\psi^{-1}:\psi(U\cap V)\to\phi(U\cap V)$ is a diffeomorphism.
- Given a function $f:U\cap V\to\mathbb{R}$, we have that $f\circ\phi^{-1}$ is smooth if and only if so is $f\circ\psi^{-1}$.
That much I've been able to prove, but does condition $(2)$ also imply condition $(1)$?
If so, how would one prove it. If not, what is a counterexample.
$\phi\circ\psi^{-1}:\psi(U\cap V)\to\phi(U\cap V)$ being a diffeomorphism is equivalent to $\phi\circ\psi^{-1}$ and $\psi\circ\phi^{-1}$ being smooth.
Now assume that 2. is satisfied. Let $\phi_i : U \cap V \to \mathbb R$ be the coordinate functions of $\phi \mid_{U \cap V} : U \cap V \to \phi(U \cap V) \subset \mathbb R^n$. The maps $\phi_i \circ \phi^{-1}$ are the coordinate functions of $\phi \circ \phi^{-1} = id$, hence they are restrictions of the canonical projections $\pi_i :\mathbb R^n \to \mathbb R$ and therefore they are smooth. By 2. the maps $\phi_i \circ \psi^{-1}$ are smooth. Thus all coordinate functions of $\phi\circ\psi^{-1}$ are smooth. We conclude that $\phi\circ\psi^{-1}$ is smooth.
Smoothness of $\psi\circ\phi^{-1}$ is proved similarly.