So, suppose that where are closed subspaces of and and are continuous functions such that then is well-defined since agree on the intersection of their domains. Also, it's continuous since if is closed it is easy to see that . But, since are continuous are closed subspaces of respectively but since they themselves are closed subspaces of it follows that so are . Thus, is the finite union of closed sets and thus closed. The conclusion follows.
This can be generalized, as I said, to a finite partition of closed sets or an arbitary partition of open ones.