Suppose $f:A\to B$ and $g:C\to D$ are smooth embeddings, $h:B\to D$ is a smooth map, and $i:A\to C$ is a continuous map such that $g(i(x))=h(f(x))$.
Then, how to show that $i$ is smooth?
An embedding f is a smooth map such that the inclusion $X\to f(X)$ is a diffeomorphism, and f(X) is a submanifold.
A n-submanifold is a subset S of M, such that for every p in S, there is a chart $x:U\to V$, with p in U, such that $x(U\cap S)=V\cap R^n\times 0$