Definition:
Let $M$ and $N$ be smooth manifolds.
Suppose $F: M\rightarrow N$ is any map. $F$ is said to be smooth at $p\in N$ if:
There exists a chart $(V,\psi)$ about $F(p)$ in $N$ and there exists a chart $(U,\phi)$ about $p$ in $M$ such that
$F(U)\subseteq V$ and
$\psi \circ F \circ \phi^{-1} : \phi(U) \rightarrow \psi(V)$ is smooth.
Problem: Let $M$ and $N$ be smooth manifolds. Let $F:M\rightarrow N$ be a map.
(1) If for every $p\in M$, there exists a neighborhood $U$ such that $F|_U$ is smooth, then $F$ is smooth.
(2) If $F$ is smooth, then its restriction to every open subset is smooth.
My attempt:
(1) Let $p\in M$. Let $U$ be a neighborhood such that $F|_U$ is smooth. Since open subsets of smooth manifolds are smooth manifolds we may discuss the smoothness of $F|_U$ . In particular, if there exists a chart $(V,\psi_1)$ of $U$ about $p$ and a chart $(V_2,\psi_2)$ about $F(p)$, such that $F(V_1)=F|_U(V_1)\subseteq V_2$, for which, $\psi_2 \circ F|_U \circ \psi_1^{-1}: \psi_1(V_1)\rightarrow \psi_2(V_2)$ is smooth, then since $V$ is open in $U$ and $U$ is open in $M$, $(V,\psi_1)$ is a chart for $M$ and $\psi_2 \circ F|_U \circ \psi_1^{-1}= \psi_2 \circ F \circ \psi_1^{-1}$, we conclude that $F$ is smooth.
(2) Suppose $F$ is smooth. Let $U$ be an open subset of $M$. I must show that $F|_U: U\rightarrow N$ is smooth. Let $p\in U$. By assumption, there exists a chart $(V_1,\psi_1)$ of $M$ about $p$, and a chart $(V_2,\psi_2)$ of $N$ about $F(p)$, for which, $F(V_1)\subseteq V_2$ and $\psi_2\circ F \circ \phi_1 ^{-1}: \psi_1(V_1)\rightarrow \psi_2(V_2)$ is smooth.Note that $(V_1\cap U,\psi_1|_{U\cap V_1})$ is a chart for $U$. This follows from the fact that $\psi_1: U\cap V_1\rightarrow \psi_1(V\cap U_1)$ is a homeomorphism and $V_1\cap U$ is open in $U$. Since $\psi_2\circ F \circ \psi_1^{-1}|_{U\cap V_1}$ is smooth (we restrict a smooth map to a smooth subset of the domain), and $F|_U(V_1\cap U)=F(V_1 \cap U)\subseteq F(V_1)\subseteq V_2$, we get: $\psi_2\circ F\circ \psi_1^{-1}|_{U\cap V_1}=\psi_2 \circ F|_U\circ \psi_1^{-1}: \psi_1(U\cap V_1) \rightarrow \psi_2(V_2)$, which is smooth.
Q.E.D
Please first tell me if my attempts are correct, then provide feedback, please.