MTH 265H "Functions of a Real Variable (Honors)"

	Time	Location	Instructor	Office Hours	E-mail
	MW 1230-1345	Hylan 201	Xuwen Chen	MW 1140-1230 @ Hylan 1018 or by appointment.	xuwenchen@rochester.edu

Syllabus: the real and complex number systems, basic topology, numerical sequences and series, continuity, differentiation, the Riemann-Stieltjes integral, sequences and series of functions.

Prerequisites: MTH 174 or MTH 200.

Textbook: Walter Rudin, Principles of Mathematical Analysis (3rd Edition), McGraw-Hill, 1976.

Course philosophy: This course is an introduction to real analysis, which is nothing but the theory of calculus. Moreover, this is a class which is heavily based on writing proofs. If you do not have familiarity with this, then you will have a hard time in this course.

We will also work on developing your independent reading skills in Mathematics. Most likely, the lectures will not be able to cover all the material you will be required to know. This is why reading assignments will be given on a weekly basis, enabling you also to ask focused questions in class and better understand the material.

This course is very challenging and requires a lot of time commitment. Proficiency will be achieved only by hard work and massive problem solving. Please take full advantage of my office hours.

Grading: Homework 30%, Midterm 30%, Final 40%.

Homework

Homework is assigned on GRADESCOPE every week on Wednesday, starting 9/4. It is due in one week. In order to ensure that assignments are graded promptly, and to discourage students from falling behind, LATE ASSIGNMENTS WILL NOT BE ACCEPTED UNDER ANY CIRCUMSTANCES. However, in recognition of the fact that unavoidable issues sometimes arise, the lowest two homework grades of each student's homework grades (including zeros for unsubmitted assignments) will be dropped when calculating final semester grades. Despite this policy, you should complete every assignment, even if you miss a deadline, because understanding the homework will help you perform well on exams.

There will be about 10 problems per homework. Each of these problems is worth 10 points.

Exams

The Midterm will be an in-class exam on 10/16/2024. Materials covered are up to 10/9/2024. So you have one week to review.

The Final is scheduled at Hylan 201, 1915-2215, 12/07/2024.

The final exam will have two parts. Part I’s score replaces the midterm score in the grade computation if Part I’s score is higher than the midterm score. The final’s score is still the sum of Part I and Part II. (That is, even you are very satisfied with your midterm score, you still need to do Part I of the final for that part of the credit of the final.)

In the exams, there is a chance that you are asked a HW problem. If your original HW solution was wrong but graded right and you provide that solution in the exam, your solution in the exam will be graded as it is supposed to be but your HW grade will stay the same.

Written Work (Only for 265HW)

For students who sign up for 265HW, one piece of written work is required. There are two ways to decide a topic to learn more and write about.

1. During the course, if you find something / topic is interesting and would like to learn more about it, send an email to ask the instructor for a related problem to base your written work on.

2. Talk to the instructor and discuss a possible topic to write on

Once a topic is decided, you have one month to submit an initial draft. The written work will be evaluated as formal written mathematics and you will be asked for revision if it is not up to the standard.

Deadline to decide a topic: 10/21/2024 to ensure enough time for the writing.

Course policies

1. We use absolute grading system. The cut-offs are 85% for A, 80% for B+, 75% for B, 70% for C+, 65% for C, 60% for D+, 55% for D. That is, if everyone gets above 85%, then everyone gets an A. (You can also choose, before taking the final, to use a system which has the minus grade: 88.3% for A, 85% for A-, 81.7% for B+, 78.3% for B, 75% for B-, 71.7% for C+, 68.3% for C, 65% for C-, 61.7% for D+, 58.3% for D, 55% for D-. Just let the instructor know before taking the final.)

2. Incomplete "I" grades are almost never given. The only justification is a documented serious medical problem or a genuine personal/family emergency. Falling behind in this course or problems with workload on other courses are not acceptable reasons.

3. If you miss the Midterm or the Final with valid excuses like illness or emergency, you must notify the instructor and provide supporting documentation which verifies your excuses as soon as possible. A make-up will be given in one week of the original final. If you miss the final without a valid excuse or supporting documentation, you will receive a score of 0 and no make-up will be given.

4. You are responsible for knowing and abiding by the University of Rochester's academic integrity code. Any violation of academic integrity will be pursued according to the specified procedures.

5. Math Dept. policy on unauthorized online resources: Any usage whatsoever of online solution sets or paid online resources (chegg.com or similar) is considered an academic honesty violation and will be reported to the Board on Academic Honesty. In particular, any assignment found to contain content which originated from such sources is subject to a minimum penalty of zero on the assignment and a full letter grade reduction at the end of the semester (e.g. a B would be reduced to a C). This applies even if the unauthorized content was obtained through indirect means (through a friend for instance) and/or the student is seemingly unaware that the content originated from such sources. If you have any questions about whether resources are acceptable, please check with your instructor.

(Tentative) Weekly schedule:

Week of	Topic	Reading assignment	Homework
8/26	Introduction, ordered sets (pages 1-4)	Fields, the real field, the extended real number system, the complex field, Euclidean spaces (pages 5-17)
9/2	Fields, the real field, the extended real number system, the complex field (pages 5-16)	Finite, countable, and uncountable sets (pages 24-30)	Homework 1 (due 9/11)
9/9	Euclidean spaces, finite, countable, and uncountable sets, metric spaces (pages 16-32)	Metric spaces, compact sets (pages 32-40)	Homework 2 (due 9/18)
9/16	Metric spaces, compact sets (pages 32-40),	Convergent sequences, subsequences, Cauchy sequences (pages 47-55)	Homework 3 (due 9/25)
9/23	convergent sequences, subsequences, Cauchy sequences (pages 47-54), monotone sequences (page 55)		Homework 4 (due 10/2)
9/30	Complete and incomplete metric spaces (pages 53-54), upper and lower limits (pages 55-57),  General series (pages 58-61, 71-72)	special sequences (pages 57-58)	Homework 5 (due 10/9)
10/7	series with nonnegative terms (pages 61-63), the root and ratio tests (pages 65-69), power series (pages 69-70), Summation by parts (pages 70-71)	Study for midterm.
10/14	In class midterm @ 10/16/2024	Addition and multiplication of series (pages 72-75), limits of functions (pages 83-84)	Homework 6 (due 10/23);
10/21	Addition and multiplication of series (pages 72-75), limits of functions (pages 83-84)	Continuous functions (pages 85-89), continuity and compactness (pages 89-93)	Homework 7 (due 10/30);
10/28	Continuous functions, continuity and compactness, continuity and connectedness (pages 85-93)		Homework 8 (due 11/6);
11/4	Discontinuities, monotonic functions (pages 94-97), differentiation (pages 103-111)	The Riemann-Stieltjes integral (pages 120-127)	Homework 9 (due 11/13);
11/11	Differentiation of vector-valued functions (pages 111-113), definition and existence of the integral (pages 120-127)	Properties of the integral, integration and differentiation, integration of vector-valued functions (pages 128-136)	Homework 10 (due 11/20);
11/18	Properties of the integral (pages 128-131)		Homework 11 (due 11/27);
11/25	Properties of the integral, integration and differentiation (pages 131-134), sequences and series of functions, uniform convergence (pages 143-148)		Homework 12 (due 12/4);
12/2	Continuity, integration, and differentiation in the context of uniform convergence	Study for the Final
12/9	TBA