Introduction to Analysis (Math420)

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

Syllabus

Textbook (free download): Jiri Lebl: Basic Analysis. Custom Pitt edition.

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.

Grades

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, 12-1pm, 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.

Academic Integrity:

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 5, 2020

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

HW2, Due Tuesday, Jan 21 in recitation

HW3, Due Tuesday, Jan 28 in recitation