Smooth maps (between manifolds) are continuous (comment in Barrett O'Neill's textbook)

Let $M$ and $N$ be smooth manifolds of dimensions $m$ and $n$, respectively, and let $\phi : M \to N$ be a smooth map. It is sufficient to show that $\phi$ is locally continuous, i.e., that every point $x \in M$ has a neighborhood $U_x$ such that $\phi\left|_{U_x}\right.$ is continuous.

Thus, let $x \in M$. Since $M$ and $N$ are smooth manifolds, there exist local coordinate systems $(U,\xi)$ at $x$ and $(V,\chi)$ at $\phi(x)$, and since $\phi$ is smooth, the coordinate expression $$\Phi = \chi \circ \phi \circ \xi^{-1} : \xi(U\cap \phi^{-1}(V)) \to \mathbb{R}^n$$ is smooth. Choose $U_x = U \cap \phi^{-1}(V)$. Then $U_x$ is a neighborhood of $x$. Since $U_x \subset \phi^{-1}(V)$, it follows that $\phi(U_x) \subset \phi(\phi^{-1}(V)) \subset V$ and therefore $$\phi\left|_{U_x}\right. = \chi^{-1} \circ \Phi \circ \xi = \chi^{-1} \circ \left(\chi \circ \phi \circ \xi^{-1}\right) \circ \xi.$$ The maps $\chi^{-1}$ and $\xi$ are homeomorphisms and $\Phi$ is a smooth map between Euclidean spaces, so these maps are all continuous. Therefore $\phi\left|_{U_x}\right.$ is the composition of continuous maps, and so is continuous. $\square$

Remarks: Usually smoothness is only defined this way for continuous maps $\phi$, so that the requirement that $U \cap \phi^{-1}(V)$ be open is automatically satisfied. More recent books such as John M. Lee's Introduction to Smooth Manifolds use the following definition of smoothness:

A map $\phi : M \to N$ between smooth manifolds is smooth if for every $x \in M$ there exist charts $(U,\xi)$ at $x$ and $(V,\chi)$ at $\phi(x)$ such that $\phi(U) \subset V$ and the coordinate expression $\Phi = \chi \circ \phi \circ \xi^{-1}$ is smooth.

Using this definition slightly shortens the proof given above, since you don't have to worry about the domains. For continuous maps, this definition implies the condition used by O'Neill.

you have to know that your coordinate charts are smooth which follows essentially from the def. of a manifold (locally homeomorphic to open euclidean sets). Then represent your map in question simply as $\eta^{-1} \eta \Phi \xi^{-1} \xi$. The middle term is cont. by def. and the outer twos by what I just said.