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?