So I am really stuck on condition (3)....For condition (1) I said does not hold since interval A is not closed ; (2) I checked the limit from the right and left at x=1 said the limit does not exist thus not continous ; (3) The intersection is the empty set, here I want to say condition (3) fails but condition (3) is the one that is giving me a hard time

Let X = A U B where A and B are closed in X. Let f : A - Y and g : B - Y be continuous. If f(a) = g(x) for every a E An B, then we can combine f and g into a continuous function h : X - Y defined by h(x) = f(x), TEA g(x), LEB Your initial post Consider X = [0, 2] and the function f : X - R defined by f(2) = X , 0Kx