Smooth map on submanifold

Question

Is the following true?

Let $M$ be a differential manifold and $f : M \to M$ be a smooth map. If $N$ is a submanifold of $M$ and $f(N)\subset N$ then the restriction $f|_N : N \to N$ is smooth.

Answer 1

It's always true that if $f\colon M\to M'$ is smooth (for any manifolds $M$ and $M'$), then the restriction of $f$ to any submanifold $N$ of $M$ is smooth. Its image is not relevant. (Of course, I assume that by "submanifold" $N$ is meant embedded submanifold.)