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Homework 8 Calculus 1
1. Prove that f : A → B is continuous if and only if f−1((a, b)) is open in A, for every open interval (a, b) ⊂ B.
2. If f : R → R satisfies the equation
f (x + y) = f (x) + f (y)
for all x, y ∈ R, and f (1) = 1,
(a) Find the values of f (x) for all x ∈ Q.
(b) If in addition f is continuous, show that f (x) = x.
3. Rudin Chapter 4, Problem 15. (See Definition 4.28 for monotonic functions.) 4. Rudin Chapter 4, Problem 23 (Just prove the first statement).
5. Prove that if f (x) is monotonic on [a, b] and satisfies the conclusion of intermediate value theorem, then f (x) is continuous.
6. Rudin Chapter 5, Problem 1. (You may assume the fact that f0 ≡ 0 ⇒ f is constant.) 7. Salas 3.1: 9, 14, 18, 35, 45, 52, 59.
