Homework 8

Review

1. A subset \$S_0\$ of a set S is dense in S if every neighborhood of a point in S contains a point in \$S_0\$. Are the rationals dense in the reals? Are rationals with powers of 2 in the denominator dense in the reals? Explain.

Connectedness

2. Prove that the integers are disconnected (directly from the definition of connected).

3. Complete problem 54 in Chapter 2 of the text. For a counterexample, look for a subset of \$\mathbb{R}^2\$.

4. Complete problem 55 in Chapter 2 of the text. In each case, provide a proof or give a counterexample.

Applications of Connectedness and Compactness

5. Prove that your height (in inches) once equaled your weight (in pounds).

6. Prove that a continuous, integer-valued function from \$\mathbb{R}\$ to \$\mathbb{R}\$ is constant. (Hint: what powerful theorem applies to this situation?)

7. Give an example demonstrating how one might use the connectedness property to prove that a function is not continuous.

8. We know that continuous functions on closed, bounded subsets of \$\mathbb{R}\$ have minimum and maximum values. Give examples of (i) a function on \$(0,1)\$ which is not continuous and has no maximum; (ii) a function on a closed but unbounded set which has no maximum; (iii) a function on a bounded but not closed set which has no maximum.