# Definition of smooth functions on a Manifold

Definition. Let $$M$$ a smooth manifold of dimension $$n$$. A funciton $$f\colon M\to \mathbb{R}$$ is said to be smooth at a point $$p$$ in $$M$$ if there is a chart $$(U,\varphi)$$ about $$p$$ in $$M$$ such that $$f\circ\varphi^{-1}$$ is smooth in $$\varphi(p)$$.

The definition of the smothness of a function $$f$$ at a point is independet of the chart $$(U,\varphi)$$, for if $$f\circ \varphi^{-1}$$ is smooth at $$\varphi(p)$$ and $$(V,\psi)$$ is any other chart about $$p$$ in $$M$$, then on $$\psi(U\cap V)$$, $$f\circ\psi^{-1}=(f\circ \varphi^{-1})\circ (\varphi\circ\psi^{-1}).$$

It is clear that $$\varphi\circ\psi^{-1}$$ is smooth in $$\psi(p)$$.

Question. Why also $$\big(f\circ\varphi^{-1}\big)$$ is smooth in $$\psi(p)?$$

We have

$$\varphi : U \to U' \subset \mathbb R^n, f \circ \varphi^{-1} : U' \to \mathbb R$$

$$\psi : V \to V' \subset \mathbb R^n, f \circ \psi^{-1} : V' \to \mathbb R$$

The transition function between $$\varphi$$ and $$\psi$$ is then

$$\varphi \circ \psi^{-1} : W' = \psi(U \cap V) \to \varphi(U \cap V) = W''.$$

$$W'$$ is an open neigborhood of $$\psi(p)$$ and $$W''$$ an open neigborhood of $$\varphi(p)$$. We get

$$(f \circ \psi^{-1}) \mid_{W'} = (f \circ \varphi^{-1}) \mid_{W''} \circ \phantom{.} (\varphi \circ \psi^{-1}).$$

Hence if $$f \circ \varphi^{-1}$$ is smooth in $$\varphi(p)$$, then so is $$(f \circ \varphi^{-1}) \mid_{W''}$$. Therefore $$(f \circ \psi^{-1}) \mid_{W'}$$ is smooth in $$\psi(p)$$ and the same is true for $$f \circ \psi^{-1}$$.

This is what has to be shown. The question whether $$\big(f\circ\varphi^{-1}\big)$$ is smooth in $$\psi(p)$$ does not make sense. In general $$\psi(p)$$ is not even contained in $$U'$$.