Let $f:\mathbb{R}^n\to\mathbb{R}^m$ be smooth, and suppose $M$ and $N$ are submanifolds of $\mathbb{R}^n$ and $\mathbb{R}^m$. Moreover, assume that $f(M) \subseteq N$. I want to prove that $f:M\to N$ is differentiable.
The outline of my proof is that since $M$ and $N$ are submanifolds $\exists$ charts $(W_1,G_1)$ at $x \in M$ and $(W_2,G_2)$ at $f(x) \in N$ and the map $G_2^{-1} \circ F \circ G_1: W_1 \to W_2$ is differentiable (because $F$ is smooth and $G_1$ and $G_2$ are homeomorphic).
Did I miss anything? I appreciate it if someone can evaluate my proof. Thanks.
You must use charts $(W_1,G_1)$ at $x \in \mathbb{R}^n$ and $(W_2,G_2)$ at $f(x) \in \mathbb{R}^m$ where
(1) $W_1$ is an open neighborhood of $x \in \mathbb{R}^n$ and $G_1 : W_1 \to U_1$ is a diffeomorphism onto an open $U_1 \subset \mathbb{R}^n$ such that $G_1(W_1 \cap M) = U_1 \cap \mathbb{R}^k \times \lbrace 0 \rbrace$.
(1) $W_2$ is an open neighborhood of $f(x) \in \mathbb{R}^m$ and $G_2 : W_2 \to U_2$ is a diffeomorphism onto an open $U_2 \subset \mathbb{R}^m$ such that $G_2(W_2 \cap N) = U_2 \cap \mathbb{R}^l \times \lbrace 0 \rbrace$.
This produces charts $G_1' : W_1' = W_1 \cap M \to U_1' \subset \mathbb{R}^k$ and $G_2' : W_2' = W_2 \cap N \to U_2' \subset \mathbb{R}^l$ which allow to deduce that $f : M \to N$ is smooth around $x \in M$.
By the way, it is known that the inclusion $i : M \to \mathbb{R}^n$ is smooth so that $f \mid_M : M \to \mathbb{R}^m$ is automatically smooth.