Arjun Krishnan
MW 12.30PM - 1:45PM Hylan 201
MATH 164 and 235, or MATH 164 and 200, or MATH 174
Real number system, continuity and uniform continuity, mean value theorems, bounded variation, Riemann-Stieltjes integral, sequences of functions.
This course is the theory behind calculus. Everything you were asked to take on faith in MTH 161-164 gets proved here, starting from the completeness of the real numbers and building up.
The course has three aims:
To develop the theory of functions of a real variable rigorously: the real number system, the topology of metric spaces, convergence of sequences and series, continuity, differentiation, the Riemann-Stieltjes integral, and sequences and series of functions.
To teach you to write proofs in real analysis. This course is heavily based on writing proofs. If you have not written proofs before, you will find this course hard; plan your time accordingly.
To teach you to read mathematics independently. Reading assignments are listed on the schedule for every week, and you are expected to have read the sections before they are lectured.
By the end of this course, you should be able to:
Read a section of Rudin on your own and reconstruct its arguments.
Write a complete, correct, and clearly organized proof from scratch, and read someone else’s proof critically enough to find the gap in it.
In terms of content you ought to be able to
State the least upper bound property and use suprema and infima correctly in arguments.
Work with the topology of metric spaces, prove statements about open, closed, compact, and connected sets. Prove and apply the Heine-Borel theorem.
Determine whether a sequence or series converges, and apply the standard convergence tests to series.
Prove statements about continuity and uniform continuity, and use the interaction of continuity with compactness and with connectedness.
Prove the mean value theorems and use them.
Construct the Riemann-Stieltjes integral, decide when a function is integrable, and prove the fundamental theorem of calculus.
Distinguish pointwise from uniform convergence, and determine when limits may be exchanged with integrals and derivatives.
| Homework | 20% |
| Midterm | 40% |
| Final | 40% |
Guaranteed grades: if you make these scores, then you are guaranteed a letter grade in the following ranges.
| Grade | Cutoff |
|---|---|
| A | 90 |
| B | 80 |
| C | 70 |
| D | 60 |
| E | below 50 |
The cutoffs may be moved down to accomodate for variation in exam difficulty. Note that this encourages collaboration: if all of you get 90s, you all get As. On the other hand, it also encourages you to use an AI to solve HW, and that’s why it’s weighted so low. However, this means that you will struggle on the exams.
Principles of Mathematical Analysis, 3rd Edition. Rudin.
Supplementary: Elementary Analysis. Ross.
The week-by-week list of topics and reading assignments is on the schedule.
Homework is assigned weekly on Friday and is due the following Friday. The problem sets themselves are on the homework page.
Each set has about ten problems. Three of them, chosen after the sets are collected, are graded and are worth 4 points each; the remaining point (lucky number 13) is awarded for a substantial attempt at every problem on the set.
I will drop your two lowest homeworks. Late homework is not accepted, which is what the two dropped scores are for: if you are ill, travelling, or simply had a bad week, that is the mechanism.
You may work with your classmates on homework. Write up your own solutions and state who you collaborated with on each problem.
The midterm is in class on Monday 10/26. The final is scheduled by the registrar during the exam period, 12/18 - 12/23. Past exams and problems will be posted on the exam page.
Each problem is scored on a four-point scale rather than by partial credit. A proof with a gap in it is not seventy percent of a proof, and the scale says so honestly:
| Score | Meaning |
|---|---|
| 4 | Perfect. Complete, correct, and clearly written. |
| 3 | Mostly correct. The right approach, carried through, with minor gaps or writing problems. |
| 2 | Many good ideas, some errors. Substantial progress, but with real errors or missing steps. |
| 0 | Not assessable. Little or no correct progress. |
If you score a 2 or a 3 on a midterm problem, you may rework it. A reworked problem rises by at most one level: a 2 can become a 3, and a 3 can become a 4.
To submit a correction, hand in both the corrected solution and one or two sentences saying what you had misunderstood the first time. Corrections without that written reflection will not be regraded. They are due one week after the graded exams are handed back in class.
The midterm is worth more to you as something you learn from than as something you are measured by. Take it seriously the first time anyway: a problem scored 0 cannot be reworked, and the final is not revisable.
If you miss the midterm with due to illness or a genuine personal emergency, notify me as soon as possible and provide documentation. In that case the final will count as your make-up, and no separate exam will be given. If you miss the midterm without a valid excuse and documentation, you will receive a 0 on it. There are no make-up exams for the final.
Incomplete grades are almost never given. The only justification is a documented serious medical problem or a genuine personal or family emergency.
All assignments and activities associated with this course must be performed in accordance with the University of Rochester’s Academic Honesty Policy.
You may work together on homework, but copying on homework or exams is not allowed, and it will be considered academic dishonesty.
Any usage whatsoever of online solution sets or paid online resources (chegg.com, chatGPT 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.
Work with your classmates (but don’t copy assignments). It is essential to not fall behind because each lecture is based on previous work. If you are having any difficulties, seek help immediately. I obviously cannot police AI use for homework, but if you do not practice solving problems, you will not do so well on the midterms. So work hard on the problems, talk to your classmates, attend office hours, and use a LLM as a last resort.
There are several avenues for you to get help and ask questions, outside of lecture:
Weekly seminar: Joshua runs a seminar once a week in which he works through problems from the current homework set and answers questions about the material from lecture. Time and location TBA. This is the single most useful hour in the week for this course, and you should treat it as part of the class rather than as optional help.
Instructor office hours or schedule an appointment to meet with your instructor.
Undergraduate Study Hall: Mondays through Thursdays 5-8pm. It’s usually somewhere on the 11th floor of Hylan.
There are other resources in the math department and at the university level.
The University of Rochester respects and welcomes students of all backgrounds and abilities. In the event you encounter any barrier(s) to full participation in this course due to the impact of a disability, please contact the Office of Disability Resources. The access coordinators in the Office of Disability Resources can meet with you to discuss the barriers you are experiencing and explain the eligibility process for establishing academic accommodations.
Office of Disability Resources (disability@rochester.edu; (585)275-9049; 1-154 Dewey Hall)
To be granted alternate testing accommodations, you (the student) must fill out forms with the Office of Disability Resources at least seven days before each and every exam. These forms are not sent “automatically.” Professors are not responsible for requesting alternative testing accommodations at the Office of Disability Resources, and they are not obligated to make any accommodations on their own.