1. The problem statement, all variables and given/known data 1) If f is a continuous mapping from a matric space X to metric space Y. A E is a subset of X. The prove that f(closure(E)) subset of closure of f(E). 2) Give an example where f(closure (E)) is a proper subset of closure of f(E). 2. Relevant equations 3. The attempt at a solution My problem is the second part, although I am unsure of my solution to part (1) as well. If E is closed then E= closure of E. Hence, for all x that belong to E , f(x) belong to Y. Hence, f(closure of E) = f(E) = set of all f(x). Now, the closure of the set of all f(x) shall be f(x) (as collection of f(x) in Y is also closed). Hence, the closure of f(E)= f(closure of (E)). Now, if E is open, then closure of E = E U E', where E' contains limit points. Now, f(closure of E) = f(E) and the set of limit points that are not in E. Where as f(E) shall be f(x) . (Kind of lost here). Also, can't think of a possible example that satisfies (2). I was thinking f(x) = 1/x , where x = (0,infinity). But this does not follow proper subset example. Please help.