ADVANCED CALCULUS, Math 127 B Homework 1 Due Friday January 16

1. Decide whether R with the following functions as proposed metrics make up a real metric space: (a) d(x, y) = |x^{2} − y^{2}| (b) d(x, y) = \frac{|x−y|}{1+|x−y|}.

2. Let Q be the set of rational numbers with the usual distance function (e.g d(x, y) = |x − y|). Is this a metric space? If yes, is the Heine-Borel theorem still true for this? Prove or disprove.

3. Let F : R^{k} → R be a continuous function. Let Z(f) = {p ∈ R^{k}|f(p) = 0}. Prove Z(f) is a closed set.

4. Exercise 21.9

5. Exercise 21.10

6. Exercise 22.1

7. Exercise 22.6