In a book I read the following -
Suppose, $M$ is a $n$-dimensional topological manifold. Each point in $M$ is in the domain of a co-ordinate map $\phi : U \rightarrow V \subset \mathbb{R^n} $. A plausible definition of a smooth function on $M$ would be to say that $f : M \rightarrow \mathbb{R}$ is smooth iff $f \circ \phi^{-1} : V \rightarrow \mathbb{R} $ is smooth in the sense of ordinary calculus.
My query is regarding the usage of $\phi$ function. Why do we have to define such a composition of functions in order to check for the smoothness of $f$. And how does the smoothness of composition function ensures the smoothness of $f$ ?