Smooth functions between manifolds

The definition of a smooth function on a manifold and a smooth path can all be subsumed in the following definition.

Definition 4.3   Let $f \colon M \to N$ be a map between manifolds. Then $f$ is called smooth if for every point $x \in M$ there are co-ordinate chart $(U, \psi)$ on $M$ and $(V, \chi)$ such that $x \in U$ and $f(U) \subset V$ and

$$\chi\circ f \circ \psi^{-1} \colon \psi(U) \to \chi(V)$$

is smooth.

Again we have the same sort of lemma:

Lemma 4.3   Let $f \colon M \to N$ be a smooth map between manifolds. Assume that there are co-ordinate chart $(U, \psi)$ on $M$ and $(V, \chi)$ such that $f(U) \subset V$. Then

$$\chi\circ f \circ \psi^{-1} \colon \psi(U) \to \chi(V)$$

is smooth.