Image of the boundary is equal to the boundary of image.

Question

In my studies of continuum mechanics, I encountered a question which I shall pose. Suppose that $f:U \to V$ is some function with domain $U\subseteq M$ and codomain $V\subseteq N$. Assume $(M,d_M)$ and $(N,d_N)$ are some metric spaces.

1. Suppose that for all $A\subseteq U$ we have $ f (\partial_M A)=\partial_N f(A)$? Then what are the implications for $f$? Should it necessarily be a homeomorphism?

2. Conversely, if $f$ is a homeomorphism, then does is imply that for all $A\subseteq U$ we have $ f (\partial_M A)=\partial_N f(A)$?