Math 104: Homework 7 (due March 18)

1. Ross exercise 19.6

2. (a) Let S be a subset of , and let f : S → and g : → be uniformly continuous functions. Prove that the composition g ∘ f : S → is uniformly continuous.

(b) Let f and g be two uniformly continuous functions from S to . Prove that f + g is uniformly continuous.

(c) Show that there exist uniformly continuous functions f and g from S to such

that the multiplication f · g is not uniformly continuous.

3. Ross exercise 20.12

4. Ross exercise 20.16

5. Ross exercise 23.5

6. Ross exercise 24.2

7. Ross exercise 24.16

8. Let f be a real-valued continuous function. Recall that for any subset S ⊆ , then f(S) is defined as {f(x) : x ∈ S}. Suppose that f(I) is open for any open interval I. Prove that f is monotonic.