Restriction of a map between manifolds is smooth?

Question

Let $f: \mathbb{R}^m \to \mathbb{R}^n$ be smooth. Suppose there is an inclusion $M \subset \mathbb{R}^m$ and $N \subset \mathbb{R}^n$. Suppose that $f(M) \subseteq N$.

Is it true that the restriction is a smooth map between manifolds $f|_M: M \to N$?

Answer 1

I guess you're assuming $f$ is $\mathscr{C}^\infty$. In that case, just observe that $f|_M=f\circ i$ where $i:M\hookrightarrow{} \mathbb{R}^m$ is the inclusion map. This is a composition of smooth maps, and hence smooth.