Assume X is a metric space, and X = $\displaystyle A\cup B $ where A and are closed subspaces. Can a boundary point of A or B be a boundary point of $\displaystyle A\cup B $ ? My guess is that it can be, that a metric space can have a boundary. But if that's so, I am misunderstanding something else: the textbook problems I am working on involve continuity on closed sets, and the textbook defined continuity of f at x_{o}requiring that for any open set in the image of f that contains f (x_{o}) there be an open set in the domain containing x_{O}s.t. the image of one open set be contained in the other. I don't see how to construct an open set in the domain that includes a boundary point of the domain.