1. Suppose that f : → is a continuous function such that f (x + y) = f (x) + f ( y) for all x, y ∈ R. Prove that there exists k ∈ such that f (x) = k x, for every x ∈ . 2. Suppose that f : ( a, b)...

1. Suppose that f :

→

is a continuous function such that f (x + y) = f (x) + f ( y) for all x, y ∈ R. Prove that there exists k ∈

such that f (x) = k x, for every x ∈
.

2. Suppose that f : ( a, b) → R is continuous and that f (r) = 0 for every rational number r ∈ ( a, b). Prove that f (x) = 0 for all x ∈ ( a, b).

3. Mark each statement True or False. Justify each answer.

(a) Let D be a compact subset of R and suppose that f : D → R is continuous. Then f (D ) is compact.

(b) Suppose that f : D → R is continuous. Then, there exists a point x 1 in D such that f (x 1) ≥ f (x) for all x ∈ D.

(c) Let D be a bounded subset of R and suppose that f : D → R is continuous. Then f (D ) is bounded.