Let X be a topological space with the property that, for every topological space Y, every function f : X Y is continuous. Prove that X has the discrete topology. (Hint: Let Y be the space X but with the discrete topology.)

Give examples of functions f and g such that f and g do not have limits at a point c, but such that both f+g and fg have limits at c.

Let (¥,r) and (¥ ,4) be two topological spaces. Prove that a function f :(X .r)——(¥ ,A) is continuous if and only if the inverse image under / of every A- closed set is a r- closed set.

Let A and B be sets. Let f : A → B be a function. Let S ⊆ A be a subset. Prove that graph(f|S) = graph(f) ∩ (S × B).

Let f : A → B be a function between sets A and B. Let X1, X2 ⊆ A and Y1, Y2 ⊆ B. a) Prove that if X1 ⊆ X2, then f(X1) ⊆ f(X2). b) Prove that if Y1 ⊆ Y2, then f -1 (Y1) ⊆ f -1(Y2)