When is a continuous map $f: M \longrightarrow N$ between smooth manifolds homotopic to a smooth one?

Question

I know there have been similar questions here, but I haven't been able to completely pin down the precise conditions on $M$ and $N$. I have seen one proof of this that uses tubular neighborhood existence, which kind of assumes that $N$ is closed.

1. Do we have to have either of them compact?

2. Can either have a boundary?

The answer will clarify a few things in my reading. Thanks.

Answer 1

This was answered in the comments: The manifold is allowed to be non-compact and have boundary. Moreover, if one knows that the map is already smooth on a closed subset we can take the approximating function to coincide on this closed subset. This is answered in Hirsch' Differential Topology.

Ps: Tubular neighborhoods also exist for non-compact embedded submanifolds. This is also in Hirsch' book.