# Question about the Definition of Smoothness at a Point

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 $$M$$.

• Can you try to look at what it says for the easiest manifold you can think of ? (e.g. $M=N=\mathbb{R}$ – Max Feb 3 at 13:48
• 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) – Minato Feb 3 at 13:57
• 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 $C^\infty$. – Max Feb 3 at 14:01
• 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! – Minato Feb 3 at 14:08
• 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 – Max Feb 3 at 14:13