1. Let A ⊆ R. We say that an element c ∈ A is isolated if there is an ε > 0 such that A ∩ (c − ε, c + ε) = {c}. (a) Show that c ∈ A is an isolated point if and only if it is not a limit point of A. (b) Show that every function f : A → R is continuous at each isolated c ∈ A. (c) Show that every function f : Z → R is continuous on its domain Z. 2. For each of the following functions f : A → R, find f(A) and hence decide whether f (equivalently its range) has an upper bound, a lower bound, a maximum or a minimum. (a) f(x) = x 3 , A = (−3, 2). (b) f(x) = x 2 , A = (−3, 2) (c) f(x) = ( x if x ∈ Q 0 if x /∈ Q , A = [0, a] where a > 0. 3. Assume f : R → R is continuous on R and let K = {x : f(x) = 0}. Show that K is a closed set. 4. Give an example of each of the following, or state that such a request is impossible. For any that are impossible, supply a short explanation for why this is the case. (a) A continuous function f : (0, 1) → R and a Cauchy sequence (xn) in (0, 1) such that (f(xn)) is not a Cauchy sequence. (b) A continuous function f : [0, 1] → R and a Cauchy sequence (xn) in [0, 1] such that (f(xn)) is not a Cauchy sequence. (c) A continuous function f : [0, 1] → R which has a maximum but no minimum. (d) A continuous bounded function f : (0, 1) → R that attains a maximum value but not a minimum value. 5. (a) Let f be a continuous real-valued function with domain (a, b). Show that if f(x) = 0 for each rational number x in (a, b), then f(x) = 0 for all x ∈ (a, b). (b) If f and g are continuous real-valued functions with domain (a, b) and f(x) = g(x) for all rational x ∈ (a, b), must f and g be the same function?