Hello forumites! I got another question (I have a lots of them, maybe I should change my textbook haha). Suppose you have a continuous real-valued function f on the metric space X and a closed set E in X. Is f(E) closed in the reals? By definition, I know that if f(A) is closed, so is the inverse image of f(A) for a set A in X if f is continuous. But if we know that E is closed, would it be correct to say that f(E) is too? Thanks a lot!