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In John M. Lee's Introduction to Smooth Manifolds (2nd edition), I have a problem of figuring out how, in practice, determine if a map is smooth.

The definition is the standard that ensures that the map is continuous. This is $f: M \to N$ is smooth at $p \in M$ iff there are admissible charts, (i.e. the charts are in the maximal atlas), $(U, \phi)$ of $M$ and $(V,\psi)$ of $N$, s.t. $p \in U$ and $f(U) \subseteq V$ and $\psi \circ f \circ \phi^{-1}$.

My issue is with the condition $f(U) \subseteq V$. It is of course important to check this condition. However, on page 37 of the book, Lee suggests that we can check if a map is smooth by just writing the map in smooth local coordinates and check that the component functions are smooth. The following examples then just checks that the coordinate representations are smooth, without explicitly stating the condition needed for the domain of the charts, i.e. $f(U) \subseteq V$. So I get the impression that this might not be necessary to check if the coordinate functions are smooth...

Related to this Lee also have proposition 2.5 (b) on page 35, which states that it is easier to check the smoothness if we know that the function $f$ is continuous, and perhaps this is somehow used in Example 2.13?

I think my two questions can be summarized as this:

  1. In the examples on page 37-38 (example 2.13) is it true that the condition $f(U) \subseteq V$ should be checked, even if this is not written out explicitly? How would one do this in practice e.g. for the map $\pi : \mathbb{R}^{n+1} \setminus \{0\} \to \mathbb{R} \mathbb{P}^n$?

  2. If we have a function $f : M \to N$, how would we know if this function is continuous? I know the standard definition of continuity (checking that the preimage of an open set is open), but this is not so easy to do in practice.

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    $\begingroup$ For your first point, if you managed to express $f$ in local coordinates, then you must implicitly have checked something like $f(U)\subseteq V$. For your second point, this might be problematic if you don't anything about your manifolds. This is true. In the setting of sub-manifolds of $\mathbb{R}^n$, you can check it the same way you'd check continuity on $\mathbb{R}^n$ if that makes you more happy. $\endgroup$ Sep 23, 2019 at 9:43

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Ok, I think I have now managed to figure this out partly with help of the comment from https://math.stackexchange.com/users/686397/woolierthanthou.

  1. I believe that you actually have to check the condition for the domain, and that the examples given are checked by directly using the definition.

  2. Checking continuity can sometimes be done directly. E.g. if we have a quotient map, which is continuous by definition.

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