Suppose $M$ and $N$ are smooth manifolds. Let $F:M \to N$ be any map. The definition of smoothness of $F$ is the following:
Def: We say that $F$ is smooth if for each $p\in M$ there exist $(U,\phi)$ smooth chart for $M$ in $p$ and $(V,\psi)$ smooth chart for $N$ such that $F(U) \subseteq V$ and $\psi^{-1} \circ F \circ \phi:\phi(U)\to \psi(V)$ is smooth in $\phi(U)$ in the sense of Ordinary Calculus.
However, if I want to give a definition of smoothness at a point, I see at least two possible ways to do so:
Let $p\in M$.
Definition 1) I say that $F$ is smooth at $p$ if there exist $(U,\phi)$ smooth chart for $M$ in $p$ and $(V,\psi)$ smooth chart for $N$ such that $F(U) \subseteq V$ and $\psi^{-1} \circ F \circ \phi:\phi(U)\to \psi(V)$ is smooth in $\phi(U)$ in the sense of Ordinary Calculus.
Definition 2) I say that $F$ is smooth at $p$ if there exist $(U,\phi)$ smooth chart for $M$ in $p$ and $(V,\psi)$ smooth chart for $N$ such that $F(U) \subseteq V$ and $\psi^{-1} \circ F \circ \phi:\phi(U)\to \psi(V)$ is smooth in $\phi(p)$ in the sense of Ordinary Calculus.
Are they equivalent? If not, which one is the correct?
I noticed that:
Pros of Definition 1)
1) $F:M \to N$ is smooth (in M, according to the "Def" above) if and only if $F$ is smooth at $p$ for each $p$ in $M$.
Cons of Definition 1)
1) Smoothness of $F$ at $p$ implies smoothness of $F$ in an entire neihborhood of $p$ (which seems a too strong requirement)
2) In Ordinary Calculus the definition of smoothness at a point does not imply the smoothness in an entire neighborhoods of that point.
Pros of Definition 2)
1) It seems more similar to the definition of smoothness at a point given in Ordinary Calculus
Cons of Definitions 2)
1) I don't know if $F:M \to N$ is smooth (in M, according to the "Def" above) if and only if $F$ is smooth at $p$ for each $p$ in $M$.