Restricting a differentiable function to a submanifold.

Question

If $f: M \longrightarrow N$ is a differentiable function between manifolds and $A$ is a submanifold of $M$, can I conclude that $f_{|_A}$ is a differentiable function? It seems that the answer should be "Yes." (we are restricting a differentiable function to the good sets for that property). The statement is clearly true in the case $A$ is open (the charts of $A$ are the charts of $M$ restricted to A), but how about in the general case?