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Loring Tu's An introduction to manifold, Proposition 6.3 asserts that:

Let $M$ be a smooth $n$-manifold and $f:M \to \Bbb R$ a real valued function on $M$. Then the following are equivalent:

(1) The function $f:M \to \Bbb R$ is $C^{\infty}$

(2) $M$ has an atlas such that for every chart $(U,\varphi)$ in the atlas $f \circ \varphi ^{-1}$ is $C^{\infty}$

But, I have a trouble with this. Since $M$ is a smooth $n$-manifold, it already has a smooth structure, and condition (1) is saying $f$ is smooth with respect to the structure of $M$. On the other hand, (2) is saying that $M$ has another atlas. Although the atlas in (2) is contained in a unique maximal atlas of $M$, it is not necessarily the same with the already given maximal atlas of $M$, isn't it?

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The atlas in $(2)$ must be compatible with the smooth structure on $M$. What $(2)$ is saying is that you don't need to use a maximal atlas, just any atlas (it cuts down on the work needed to check things if there's an atlas with a finite number of charts).

The content in $(2)$ is that if $f$ is smooth with respect to an atlas $\mathcal{A}$, then it is also smooth with respect to the maximal atlas containing $\mathcal{A}$. The content is the extension to the charts of the maximal atlas not in $\mathcal{A}$.

For example, on a circle (with the standard atlas) you only need to check that a function $f$ is smooth with respect to the upper, lower, left, and right semicircles to conclude that $f$ is smooth instead of checking that its smooth with respect to all charts in a maximal atlas.

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  • $\begingroup$ Thanks this is exactly what I wanted $\endgroup$ – probably123 Jan 9 '20 at 13:10

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