## Description

1. [10 points] Given S = Q (that is, the set of all rational numbers), find int S, bd S, and classify S as open, closed, neither, or both. Justify your answers by showing full proofs. 2. [10 points] Find the maximum and supremum of the set S = sup{x ∈ Q : x < π}, and justify your answers by showing full proofs. 3. [10 points] Determine whether or not the following limits exist and justify your answers by showing full proofs. (a) limx→0 x 2 sin 1 x . (b) limx→0 sin 1 x . 4. [10 points] Let f : A → B be uniformly continuous on A and g : B → C be uniformly continuous on B. Prove that g ◦ f : A → C is uniformly continuous on A. 5. [10 points] Suppose that f ′ (0) exists and that f(x + y) = f(x) · f(y) for all x, y ∈ R. Prove that f is differentiable at any c ∈ R. 6. [10 points] Let f and g be differentiable on R. Suppose that f(0) = g(0) and that f ′ (x) ≤ g ′ (x) for all x ≥ 0. Show that f(x) ≤ g(x) for all x ≥ 0. (Hint: apply Mean Value Theorem on f − g). 7. [10 points] Let f(x) = x 3 for x ∈ [0, 1]. Given n ∈ N, consider the partition Pn = {0, 1 n , 2 n , ..., n−1 n , 1}. Find L(f, Pn), U(f, Pn), L(f), and U(f). 8. [10 points] Let S = {s1, s2, ..., sk} be a finite subset of [a, b]. Suppose that f is a bounded function on [a, b] such that f(x) = 0 if x /∈ S. Show that f is integrable and that ∫ b a f = 0.

### Unformatted Attachment Preview

Purchase answer to see full attachment

## Explanation & Answer

Attached. Please let me know if you have any questions or need revisions.

1·

G-,ven. S = ©. .

we doltm.

of

~ °g ;_ 11 .'Yn0..X t'Y1\..LL.'Y't'\,

At f--h--s+ obSe-Yve....