Introduction to Analysis (Math420)

This webpage is for the Spring 2020 course at the University of Pittsburgh.

All class activities are online. Homework is to be submitted digitally. Office hours only via skype or similar

Syllabus

I will post lecture notes here: lecture notes

Prerequisites

The course concerns the One-Variable Calculus. Prerequisite is Math 413: Introduction to Theoretical Mathematics. If you do not feel comfortable with the prerequisite material, please contact the instructor in the beginning of the course.

Homework

Homework will be assigned each Monday (starting in the second week of the semester), and it will be due on Tuesday the following week during the recitations. Late homework will not be accepted. The solution of each exercise will be evaluated in the scale 0-5 points, taking into account the correctness, clarity and neatness of presentation.

Your final grade depends on your performance on the final exam as well as on your total grade. Grades will be based on homework (35%), two midterms (15% + 15%) and the final (35%). There will be no make up midterm exams. If you miss the midterm exam for a documented medical reason, your grade on it will be the prorated grade of your final exam. Incompletes will almost never be given, and only for cases of extreme personal tragedy.

This course fulfills requirements for the following majors:

• Bachelor of Science in Mathematics
• Bachelor of Science in Applied Mathematics
• Bachelor of Science in Mathematics-Economics
• Bachelor of Science in Mathematical Biology

Core topics

1. The Bolzano-Weierstrass Theorem; Cauchy sequences; Cauchy completeness of the real numbers.
2. Real-valued functions on an interval: limits and continuity.
3. Intermediate Value Theorem; Max-Min Theorem.
4. Uniform continuity; continuous functions on a closed and bounded interval are uniformly continuous.
5. Differentiable functions.
6. Interior Extremum Theorem, Rolle’s Theorem, Mean Value Theorem.
7. Taylor’s Theorem and Taylor Series.
8. The Riemann Integral on a closed and bounded interval.
9. The Fundamental Theorem of Calculus.
10. Definition and examples of pointwise and uniformly convergent sequences of functions.
11. Continuity of uniform limits of continuous functions.
12. Interchange of uniform limits and integration.
13. Interchange of limits with differentiation.
14. The M-test for uniform convergence of series.
15. Application to power series.

Class hours

Tuesdays, Thursdays, 4:00-5:15pm, Benedum 1045

Office hours

Thursdays, 1-2pm, Thackeray Hall, Room 410

Further office hours available upon request by email: armin@pitt.edu

Teaching Assistant

Teaching Assistant: Dongyu Wu.

Recitations: Tuesdays, 5:30-6:15PM, PUBHL A216

Disability Resource Services:

If you have a disability for which you are or may be requesting an accommodation, you are encouraged to contact both your instructor and the Office of Disability Resources and Services, 216 William Pitt Union (412) 624-7890 as early as possible in the term.

Cheating/plagiarism will not be tolerated. Students suspected of violating the University of Pittsburgh Policy on Academic Integrity will incur a minimum sanction of a zero score for the quiz, exam or paper in question. Additional sanctions may be imposed, depending on the severity of the infraction.

On homework, you may work with other students or use library resources, but each student must write up his or her solutions independently. Copying solutions from other students will be considered cheating and handled accordingly.

Midterms

• Midterm 1: February 11, 2020
• Midterm 2: March 16, 2020

The midterm contains all theorems (incl. proofs), definitions of the lecture, as well as the homework problems.

Homework Assignments

There is a 3% bonus on your final exam, if you write all your homewerks in Latex.

HW1, Due Tuesday, Jan 14 in recitation

all references from the textbook

• pg. 34, problem 1.2.10
• pg. 38, problem 1.3.1
• pg. 54, problem 2.1.12
• pg. 64, problem 2.2.9 (Hint: consider using Lemma 2.1.12)

• week 1 solutions

Midterm 1, Due Monday, February 17th, 2pm

This is a takehome exam, which is still an exam; so you must work by yourself. Any violations of this rule will be consider cheating and will be reported to the Dietrich School. You are only allowed to talk to the instructor or the TA about the exam, but you can use the book, the lecture notes, and your own notes from class and exercises.

The Midterm is due Monday, February 17th, 2pm.

Until then you have to either

• send the solutions to me by email: armin@pitt.edu
• leave the solutions in my mailbox (in Thackeray 301)
• have given me the solutions in person.

HW9, Due Tuesday, Mar 24 in gradescope

this homework is fun, but is optional (whatever you score will replace your lowest score of the weekly assignments)

Midterm 2, Due Monday, March 30th, 2pm

This is a takehome exam, which is still an exam; so you must work by yourself. Any violations of this rule will be consider cheating and will be reported to the Dietrich School. You are only allowed to talk to the instructor or the TA about the exam, but you can use the book, the lecture notes, and your own notes from class and exercises.

• one problem per page please
• pdf please: if you submit your handwritten exam, you can use any camscanner app for your phone
• if you use latex, you can use the latex template

Instead of HW 11: Suggested Problems

There is no new homework assignment. Instead, there are some suggested problems to try.
Solutions will be provided on Friday, April 17.

6.1.7, 6.2.2, 6.2.3, 6.2.4, 6.5.5 - solutions

Final exam, Due Wednesday, April 22nd, 11:55pm

This is a takehome exam, which is still an exam; so you must work by yourself. Any violations of this rule will be consider cheating and will be reported to the Dietrich School. You are only allowed to talk to the instructor or the TA about the exam, but you can use the book, the lecture notes, and your own notes from class and exercises.