[proofplan]
We verify the smoothness criterion in arbitrary charts. Around a fixed point $p\in M$, the hypothesis gives one pair of charts in which $F$ has a smooth coordinate representation. If we replace these charts by any other source and target charts near $p$ and $F(p)$, the new coordinate representation is obtained by composing the assumed smooth coordinate representation with smooth transition maps. Since smoothness for Euclidean maps is preserved under restriction and composition, the coordinate representation is smooth near the chosen point, and the local definition of smoothness on manifolds gives the result.
[/proofplan]
[step:Fix a point and choose the charts supplied by the hypothesis]
Let $p\in M$ be arbitrary. By hypothesis, choose a smooth chart $(U,\varphi)$ of $M$ and a smooth chart $(V,\psi)$ of $N$ such that $p\in U$, $F(p)\in V$, $F(U)\subset V$, and the map
\begin{align*}
G:\varphi(U)\to\psi(V),\quad x\mapsto \psi(F(\varphi^{-1}(x)))
\end{align*}
is smooth.
We will show that $F$ is smooth at $p$. Since $p$ was arbitrary, this will prove that $F$ is smooth on all of $M$.
[/step]
[step:Rewrite an arbitrary coordinate representation using transition maps]
Let $(A,\alpha)$ be any smooth chart of $M$ with $p\in A$, and let $(B,\beta)$ be any smooth chart of $N$ with $F(p)\in B$. Define
\begin{align*}
W:=A\cap U\cap F^{-1}(B\cap V).
\end{align*}
Because $A$ and $U$ are open in $M$, because $B\cap V$ is open in $N$, and because $F$ is continuous, the set $W$ is an open neighbourhood of $p$ in $M$.
On the [open set](/page/Open%20Set) $\alpha(W)\subset\alpha(A)$, the coordinate representation of $F$ in the charts $(A,\alpha)$ and $(B,\beta)$ is
\begin{align*}
\beta\circ F\circ\alpha^{-1}:\alpha(W)\to\beta(B),\quad y\mapsto \beta(F(\alpha^{-1}(y))).
\end{align*}
For every $y\in\alpha(W)$, set $x=(\varphi\circ\alpha^{-1})(y)\in\varphi(U)$. Since $\alpha^{-1}(y)\in W\subset F^{-1}(B\cap V)$, the point $F(\alpha^{-1}(y))$ lies in $B\cap V$. Therefore the expression below is well-defined, and we have
\begin{align*}
\beta\circ F\circ\alpha^{-1}=(\beta\circ\psi^{-1})\circ(\psi\circ F\circ\varphi^{-1})\circ(\varphi\circ\alpha^{-1})
\end{align*}
on $\alpha(W)$.
[guided]
The purpose of this step is to compare two different coordinate descriptions of the same map. The hypothesis gives smoothness only in the special charts $(U,\varphi)$ and $(V,\psi)$, while the definition of smoothness may ask us to test $F$ in other charts $(A,\alpha)$ and $(B,\beta)$.
We first choose a domain on which all relevant expressions make sense. Define
\begin{align*}
W:=A\cap U\cap F^{-1}(B\cap V).
\end{align*}
This set contains $p$ because $p\in A\cap U$ and $F(p)\in B\cap V$. It is open because $A$ and $U$ are open chart domains, $B\cap V$ is open in $N$, and the continuity of $F$ implies that $F^{-1}(B\cap V)$ is open in $M$.
Now take $y\in\alpha(W)$. Then $\alpha^{-1}(y)\in W\subset U$, so $\varphi(\alpha^{-1}(y))$ is defined. Also $\alpha^{-1}(y)\in W\subset F^{-1}(B\cap V)$, so $F(\alpha^{-1}(y))\in B\cap V$, and both $\psi(F(\alpha^{-1}(y)))$ and $\beta(F(\alpha^{-1}(y)))$ are defined. Hence the following composition is meaningful:
\begin{align*}
(\beta\circ\psi^{-1})\circ(\psi\circ F\circ\varphi^{-1})\circ(\varphi\circ\alpha^{-1}).
\end{align*}
Evaluating it at $y$ gives
\begin{align*}
(\beta\circ\psi^{-1})(\psi(F(\varphi^{-1}(\varphi(\alpha^{-1}(y)))))).
\end{align*}
Since $\varphi^{-1}(\varphi(\alpha^{-1}(y)))=\alpha^{-1}(y)$, this equals
\begin{align*}
(\beta\circ\psi^{-1})(\psi(F(\alpha^{-1}(y))))=\beta(F(\alpha^{-1}(y))).
\end{align*}
Thus on $\alpha(W)$,
\begin{align*}
\beta\circ F\circ\alpha^{-1}=(\beta\circ\psi^{-1})\circ(\psi\circ F\circ\varphi^{-1})\circ(\varphi\circ\alpha^{-1}).
\end{align*}
This identity is the mechanism by which smoothness transfers from the chosen charts to arbitrary compatible charts.
[/guided]
[/step]
[step:Use smooth transition maps and Euclidean composition to get local smoothness]
The map
\begin{align*}
\varphi\circ\alpha^{-1}:\alpha(A\cap U)\to\varphi(A\cap U)
\end{align*}
is smooth because $(A,\alpha)$ and $(U,\varphi)$ are compatible smooth charts on $M$. The map
\begin{align*}
\beta\circ\psi^{-1}:\psi(B\cap V)\to\beta(B\cap V)
\end{align*}
is smooth because $(B,\beta)$ and $(V,\psi)$ are compatible smooth charts on $N$. The middle map $G=\psi\circ F\circ\varphi^{-1}$ is smooth by the hypothesis.
Restricting these smooth maps to the relevant open subsets and composing them, the Euclidean composition rule for smooth maps implies that
\begin{align*}
\beta\circ F\circ\alpha^{-1}:\alpha(W)\to\beta(B)
\end{align*}
is smooth. Thus, in the arbitrary charts $(A,\alpha)$ and $(B,\beta)$, the coordinate representation of $F$ is smooth on an open neighbourhood of $\alpha(p)$.
[/step]
[step:Conclude smoothness at every point]
We have shown that for the arbitrary point $p\in M$ and arbitrary smooth charts $(A,\alpha)$ around $p$ and $(B,\beta)$ around $F(p)$, the coordinate representation $\beta\circ F\circ\alpha^{-1}$ is smooth on an open neighbourhood of $\alpha(p)$. By the definition of a smooth map between smooth manifolds, $F$ is smooth at $p$.
Since $p\in M$ was arbitrary, $F$ is smooth at every point of $M$. Therefore $F:M\to N$ is smooth.
[/step]
