Homework 7. Due on Wednesday, November 13

1. Show that every polynomial of odd degree with real coeffients has at least one real root.

2. Let f be continuous on the interval [0, 1] to ℝ and such that f(0) = f(1). Prove that there exists a point c in [0, \frac{1}{2}] such that f(c) = f(c+ \frac{1}{2}). Conclude that there are, at any time, antipodal points on the earth’s equator that have the same temperature.

3. Let I := [a, b], and let f : I → ℝ be continuous on I and assume that f(a) < 0, f(b) > 0. Let S := {x ∈ I : f(x) < 0}, and let c := supS. Prove that f(c) = 0.

4. Prove that f(x) = \frac{1}{x^{2}} is uniformly continous on the set A := [a, ∞) but not on the set B := (0, ∞).

5. Problem 8 on page 99 of Rudin.

6. Problem 18 on page 100 of Rudin.

7. Problem 21 on page 101 of Rudin. (Assume the metric space X is ℝ, and K is a closed, bounded set in ℝ).

8. Problem 23 on page 101 of Rudin.