Suppose (X, d) is a compact metric space and f: X → R is continuous. Prove that f is bounded above and its supremum is attained at some point in X

Give an example of a function f where it does not have a global maximum on a closed interval [a, b]. (Hint: f cannot be continuous, because Theorem 4.2 guarantees a continuous function to have a global maximum on a close interval.)

Let (X, d) be a metric space and let f, g : X → R be two continuous functions. (a) Prove that the set {x ∈ X : f(x) < g(x)} is open. (b) Prove that if the set {x ∈ X : f(x) ≤ g(x)} is dense in X, then f(x) ≤ g(x) for all x ∈ X.

Let (X, dx), and (Y, dy) be metric spaces, and suppose that f : X → Y is continuous on X, and X is compact. (a) Show that f(X) is compact. (b) Show that if f is bijective then f-¹ is continuous as well.