I am currently working through Nakahara's book, "Geometry, Topology and Physics", and have reached the stage at looking at calculus on manifolds. In the book he states that "The differentiability of a function $f:M\rightarrow N$ is independent of the coordinate chart that we use". He shows this is true in $M$ and then leaves it as an exercise to show that it is also true in $N$. Here is my attempt, please could someone tell me if what I've done is correct, and if not, what the correct way is?
Let $f:M\rightarrow N$ be a map from an $m$-dimensional manifold to an $n$-dimensional manifold $N$. A point $p\in M$ is mapped to a point $f(p)\in N$, namely $f:p\mapsto f(p)$. Take a chart $(U,\phi)$ on $M$ and $(V,\psi)$ on $N$, where $p\in U$ and $f(p)\in V$. Then $f$ has the following coordinate presentation: $$\psi\circ f\circ\phi^{-1}:\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$$ If we write $\phi (p)=\lbrace x^{\mu}\rbrace =x$ and $\psi(f(p))=\lbrace y^{\alpha}\rbrace = y$, then $y=\psi\circ f\circ \phi^{-1} (x)$ is a vector-valued function of $m$ variables.
The differentiability of $f$ is independent of the coordinate chart in $N$. Indeed, let $(V_{1},\psi_{1})$ and $(V_{2},\psi_{2})$ be two overlapping charts in $N$. Take a point $q=f(p)\in V_{1}\cap V_{2}$, whose coordinates by $\psi_{1}$ are $\lbrace y_{1}^{\mu}\rbrace$, while those $\psi_{2}$ are $\lbrace y_{2}^{\nu}\rbrace$. When expressed in terms of $\lbrace y_{1}^{\mu}\rbrace$, $f$ takes the form $$y_{1} =\psi_{1}\circ f\circ\phi^{-1}$$ while in $\lbrace y_{2}^{\nu}\rbrace$, it takes the form $$y_{2} =\psi_{2}\circ f\circ\phi^{-1} =\psi_{2}\circ\left(\psi^{-1}_{1}\circ\psi_{1}\right)\circ f\circ\phi^{-1}= \left(\psi_{2}\circ\psi^{-1}_{1}\right)\circ\psi_{1}\circ f\circ\phi^{-1}$$ Now, by definition, $\psi_{2}\circ\psi^{-1}_{1}$ is $C^{\infty}$, and therefore if $\psi_{1}\circ f\circ\phi^{-1}$ is $C^{\infty}$ with respect to $y_{1}^{\mu}$ and $\psi_{2}\circ\psi^{-1}_{1}$ is $C^{\infty}$ with respect to $y_{2}^{\nu}$, then $\psi_{2}\circ f\circ\phi^{-1}=\left(\psi_{2}\circ\psi^{-1}_{1}\right)\circ\psi_{1}\circ f\circ\phi^{-1}$ is also $C^{\infty}$ with respect to $y_{2}^{\nu}$.
Would this be correct?