Math 554 Section 001 (MTWTh 10:30--12:45, in LeConte 303B) Analysis I

Summer I 2010

10:30 AM - 12:45 PM, with a 15 minute break

Office: LeConte 406

Phone: 7-5134

Email: nyikos @ math.sc.edu

Office hours:
1:30-3:00 MTWTh

The course grade is based 40% on homework and quizzes, 20% on a midterm test, and 40% on the final exam. The final exam is cumulative.

The midterm and the final will include questions about definitions, and you will also be expected to prove a number of things on these two tests, but only things whose proofs may be found in the textbook. Moreover, unless the proof was gone over in class (or an alternative proof provided in class) you will also have the option of proving something that was proven both in class and in the textbook.

The homework, on the other hand, includes problems where you have to come up with your own proofs. Except for very easy proofs, you have two chances to come up with the proofs, and I will provide some hints to make it easier for you the second time around.

This course covers parts of Chapters 1 through 6 in Introduction to Real Analysis, 2nd edition, by Manfred Stoll, with emphasis on Chapters 2 and 4. Some material from each of the following sections will be covered:
1.1 through 1.5
All sections of Chapter 2
3.1
4.1, 4.2
5.1, 5.2
6.1
Excerpts from other sections are covered as time permits.

The objectives of this course are to arrive at a a command of the basic concepts of set theory and real analysis up through integration, and the ability to prove statements in set theory and real analysis on the level appropriate to the course. Among the concepts you are expected to master are:

• the set theoretic operations of union, intersection, complementation, and cartesian product;
• functions and their inverses (if any), image and preimage of sets under functions;
• ordered fields, suprema, infima, completeness and Archimedean order of the real line;
• open and closed subsets of the real line, and the interiors of arbitrary subsets;
• boundedness of sets, sequences, and functions;
• limits of sequences, series, and functions including left and right limits, limits at infinity and infinite limits;
• continuity, differentiability and integrability of real-valued functions of a real variable.

Starred problems are not graded the first time; they are returned with hints if they haven't been done correctly, and grading only takes place the second time.

Homework due Thursday, June 3:>
Section 1.1, finish 1(a)(b) [this means: those parts not covered in class] and the parts of 2(a)(b)(c) not given in back of the book; also do 7c*
Section 1.2, do 2 (b), 3 (b) (c) and 5.

Homework due Tuesday, June 8:
Section 1.2, 6df and 8*
Section 1.3, 1c, 2d, 5a*

The final exam in this course is on Thursday, July 1, at the usual class time and place.

The last homework assignment is due on Monday, June 28, and is as follows:
Section 4.1, 8b and 17bd
Section 4.2, 1b and 10 (except for finding the range)
Section 5.1, 8bd
In doing 8b, do at least three steps and if a step uses a fact about limits (either a part of Theorem 4.1.6 or 4.1.2 (a) or some part of exercise 2) then specify which part of which theorem or exercise is used.
In doing 1d, "determine" means that if the function is continuous at , find the limit at 2, give the function value at 2, and use these two facts to explain why the function is either continuous or discontinuous at 2.