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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 $M$.

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  • $\begingroup$ Can you try to look at what it says for the easiest manifold you can think of ? (e.g. $M=N=\mathbb{R}$ $\endgroup$ – Max Feb 3 at 13:48
  • $\begingroup$ I was thinking if a map $f:\mathbb{R}\to \mathbb{R}$ which is smooth at a point $x_0$ is also smooth in a neighborhood of $x_0$ but I'm not able to give an answer. If the answer is not, then i think the correct definition of smoothness at a point for a manifold should be Definition 2) $\endgroup$ – Minato Feb 3 at 13:57
  • $\begingroup$ Well imagine a map that is more and more differentiable, the closer you get to $0$; for instance it would be $C^1$ on $]-1,1[$, $C^2$ on $]-\frac{1}{2},\frac{1}{2}[$ but not at $\frac{1}{2}, -\frac{1}{2}$, $C^3$ as you get closed but not on the full interval and so on. Then it will be $C^\infty$ at $0$, but there is no neighbourhood of $0$ at which it is $C^\infty$. $\endgroup$ – Max Feb 3 at 14:01
  • $\begingroup$ Wonderful idea, even though now I have to think at a "concrete"example. But now, is it the case that a map $F:M \to N$ which is smooth at $p$ for each $p \in M$ (according to Definition 2) is also smooth in $M$ according to "Def"? Because in Oardinary Clalculus we have that property! $\endgroup$ – Minato Feb 3 at 14:08
  • $\begingroup$ Well you can try to prove that def 1) doesn't depend on the choice of the chart as long as it's small enough; and use that to prove it $\endgroup$ – Max Feb 3 at 14:13

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