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.

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