## 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.