Practice Test 2

There are some answers in the postscript and acrobat versions.

This test is closed book, no books, papers or notes are permitted. You may use theorems, lemmas and propositions from the class and book. Note that where is mentioned below the standard metric is assumed.

There are 5 questions on the actual test, I think they are mostly easier than these ones.

1. Consider the function defined by

Show carefully, using results from class, that any monotonic increasing function which is continuous at is Riemann-Stieltjes integrable with respect to
2. Let be a continuous function on Explain whether each of the following statements is always true, with brief but precise reasoning.
1. The function is well defined.
2. The function is continuous.
3. The function is decreasing.
4. The function is uniformly continuous.
5. The function is differentiable.
6. The derivative on

3. If is differntiable and satisfies show that there is a point where

4. If is a strictly positive continuous function on meaning show that is continuous.

5. (This is basically Rudin Problem 4.14)

Let be continuous.

1. State why the the map from to is continuous.
2. Using this, or otherwise, show that is closed and is open.
3. Show that is not empty.
4. Suppose that for all and conclude that is open in and that
5. Conclude from this, or otherwise, that there must in fact be a point such that

6. Consider the function

for
1. Explain why is differentiable.
2. Compute
3. Show that there exists such that for
4. Show that there must exist a point with and

Richard B. Melrose 2004-05-18