MAT 319: Introduction to Analysis, Fall 2021

Dept of Mathematics , Stony Brook University

MAT 319 and 320

MAT 319 (Foundations of Analysis) and MAT 320 are taught as a single lecture until the first midterm. Based on the results of the first midterm, students may switch into MAT 320, which will proceed at a faster pace than MAT 320. Professor Martens will teach MAT 319, and the two of us will share lectures up until the first midterm.


MAT 319 Instructor

        Prof. Marco Martens,
        Dept. Phone: (631)-632-4893
        Dept. FAX: (631)-632-7631
        my homepage

MAT 320 Instructor

        Prof. Christopher Bishop,
        Dept. Phone: (631)-632-8290
        Dept. FAX: (631)-632-7631
        my homepage

Course Summary

This is a course in real analysis or roughly "calculus with proofs". We will start by recalling set notation, the principle of induction, and the basic properties of finite and infinite sets. We then discuss the axioms of the real numbers, and learn to rigorously derive various familiar properties. When then study sequences and limits. Around this point MAT 319 and 320 will separate. In MAT 320 will will then discuss continuity, differentiation, Riemann integration, sequences of functions and infinite series. If time permits, we will discuss basic point set topology, metric spaces, the generalized Riemann integral and the Lebesgue integral.

Time and Place

For the first few weeks, MAT 319 and MAT 320 both meet Tuesdays and Thursdays 9:45am-11:05am in Stony Brook Union room 103-02. After the classes split MAT will continue in that room and MAT 320 will meet in Mathematics P-131 (this room is directly behind the elevators on the PL level of the Math Tower). Recitations meet twice a week as follows:
        MAT 319 R01 82270 MW 11:45am-12:40pm Earth-Space Science 181
        MAT 319 R02 88825 MW 11:45am-12:40pm Frey Hall 309
        MAT 319 R03 94829 MW 11:45am-12:40pm Socbehav Sci N310
        MAT 320 R01 81315 MW 11:45am-12:40pm Physics P116
Note that all recitations meet at the same time.

Office Hours

Office Hours MAT319: TuTh 11:30-12:30 or by appointment


The textbook is "Introduction to Real Analysis, 4th Edition" by Robert Bartle and Donald Sherbert.


Grades will be based on weekly problem sets, two midterms and a final exam. Each component will count for 25% of the total grade.

Final Exam

The final exam is Tuesday, December 14, 8:00am-10:45am. Location will be be announced later. Possibly all final exams will be online.


        Blackboard is the Stony Brook University class management system. Your homework, quiz and exam grades will be posted here. Letter grades for the course are posted in the Solar System. I will occasionally post announcements in Blackboard; you should receive email notifications whenever this occurs.

Solar System

        Solar System is the Stony Brook University administrative management system (registration, bills,...). It is not used for classes, except to post letter grades at the end of the semester..

Stony Brook Gmail

        Check your email here.

Stony Brook Virtual SINC Site

        The Virtual Sinc Site gives you access to various software packages on a university license, such as Mathematica and Matlab. I don't plan to use any of these in this class, but this resource may be helpful in other classes. Using the virtual Sinc Site requires downloading the Citrix receiver software (you will be prompted).

Important University Dates

    Link to university academic calendars, including final exam calendars.

    First day of classes: Monday August 23, 2021.
    Labor Day, no classes: Monday Sept 6, 2021.
    Fall break, No classes Mon Oct 11 and Tue Oct 12
    Thanksgiving break: Wed November 24 to Sunday November 28, 2021.
    Last day of classes: Monday December 6, 2021.
    Reading day: Tuesday December 7, 2021.
    Finals: Wednesday December 8 to Thursday December 16, 2021.
    MAT 320 Final Exam: 8:00am-10:45am Tuesday, Dec 14, 2021
    Commencement: Friday December 17, 2021

University deadlines

See the following page for deadlines for things like withdrawing from classes without penalty, applying for P/NC, changing major, ... University deadlines

Pass/No Credit

This policy allows you to set a threshold so that if you score above the threshold in a class you get a that grade on your transcript, and otherwise you get a P (for pass) or NC (no credit), neither of which affects your GPA. For example, if you set the threshold at C and if you get a C- or a D you will get a P in the course (which means it won't count towards major requirements, but also won't affect your GPA). A grade of F gives an NC (also won't affect your GPA), and any grade equal to or higher than your threshold will count as usual. Generally, only one course per semester may be designated P/NC. Check the Bulletin for precise dates and requirements; your major may also have rules about which classes these may be used for. SBU G/P/NC page.

Problem sets

There will be a problem set due each week (except the first week of class and midterm weeks; sections covered the week before a midterm are due the week after the midterm). These should be handed in at the first recitation of the week. Problems will be taken from the textbook and are listed in the lecture schedule below. Problems from a section are due in recitation the week after that section is covered in lecture.

Tentative Lecture Schedule

Week 1, Aug 23 - Aug 27
        Topics covered:
        1.1 Sets and functions (problems 6, 14, 18, 21)
        1.2 Induction (problems 7, 11, 17)
        1.3 Finite and infinite sets (problems 4, 12, 13)

Week 2, Aug 30 - Sept 3
        Topics covered:
        2.1 Algebraic and order axioms (problems 4, 8, 17, 19)
        2.2 Absolute value and the real line (problems 2, 17, 18)
        2.3 The completeness property (problems 10, 12,13)

Week 3, Sept 6 - Sept 10
        Topics covered:
        2.4 Applications of the Supremum property (problems 5, 7, 11, 19)
        2.5 Intervals (problems 8, 12)
        3.1 Sequences and their limits (problems 4, 5, 11, 17)

Week 4, Sept 13 - Sept 17
        Topics covered:
        3.2 Limit theorems (problems 2, 7, 15, 20)
        3.3 Monotone sequences (problems 2, 7, 9)
        3.4 Bolzano-Weierstrass theorem (problems 6, 9, 18)

Week 5, Sept 20- Sept 24:
        Topics covered:
        3.5 Monotone sequences (problems 4, 6, 11)
        3.6 Properly divergence sequences (problems 2, 5, 10)
        3.7 Introduction to infinite series (problems 5, 11, 15, 17)

Week 6, Sept 27 - Oct 1
        Topics covered:
        Review of Chapters 1-3
        Midterm 1

Week 7, Oct 4 - Oct 8
        Topics covered:
        4.1 Limits of functions (problems 6, 13, 16)
        4.2 Limit Theorems (problems 8, 12, 14)
        4.3 Extensions of the limit concept (problems 7, 11)
        5.1 Continuous functions (problems 3, 6, 12, 14, 15)
        5.2 Combinations of continuous functions (problems 3, 8, 11, 14)

Week 8, Oct 11 - Oct 15, Fall break, no classes Oct 11-12
        Topics covered:
        5.3 Continuous functions on intervals (problems 6, 13, 18)
        5.4 Uniform continuity (problems 2, 6, 15, 16)

Week 9, Oct 18 - Oct 22
        Topics covered:
        5.5 Continuity and gauges (no problems)
        5.6 Monotone and inverse functions (problems 2, 9, 10, 12, 13)
        6.1 The deriviative (problems 7, 9, 13, 17)
        6.2 The mean value theorem (problems 8, 11, 12, 13, 15)
        6.4 Taylor's theorem (problems 8, 12, 16)

Week 10, Oct 25 - Oct 29
        Topics covered:
        7.1 The Riemann integral (problems 7, 8)
        7.2 Riemann integrable functions (problems 6 (give an example), 8, 9, 15)
        Appendix C, The Riemann and Lebesgue criteria
        The fundamental theorem (problems 8, 14, 16, 21)

Week 11, Nov 1 - Nov 5
        Topics covered:
        8.1 Pointwise and uniform convergence (problems 12, 19, 21,24)
        8.2 Interchange of limits (problems 3, 14, 17, 18)
        8.3 Exponential and logarithmic functions (problems 4, 9)
        8.4 The trigonometric functions (no problems)

Week 12, Nov 8 - Now 12
        Topics covered:
        9.1 Absolute conevergence (problems 2, 7, 8, 15, 16)
        9.2 Tests for absolute convergence (problems 15, 17, 19)
        9.3 Tests for non-absolute convegence (problems 6, 9, 10, 12)
        9.4 Series of functions (problems 2, 3, 11, 14, 15)

Week 13, Nov 15 - Nov 19
        Topics covered:
        Review of Chapters 4-9
        Midterm 2

Week 14, Nov 22 - Nov 26
        Thanksgiving break, no class Thursday.

Week 15, Nov 29 - Dec 3 Last week of classes. Some recitations meet Mon Dec 6.
        Topics covered:
        11.1 Open and closed sets (problems 10, 11, 16}
        11.2 Compact sets (problems 4, 9, 11)
        11.3 Continuous functions (problems 2, 7, 10)
        11.4 Metric spaces (problems 8, 9, 10)

Final Exam on Tuesday Dec 14, 8:00am-10:45pm.

Topics in the history of Calculus

Below are some reading about the history of calculus that may be of interest.
        Wikipedia page on calculus.
        Wikipedia page on Issac Newton.
        Wikipedia page on Newton-Leibniz controversy.
        Wikipedia page on the discovery of the planet Neptune (using only mathematics).
        Wikipedia page on Gauss.
        Wikipedia page on Pappus.

Rankings of math departments - 2020

        The 2021 Shanghai Ranking of mathematics departments around the world. Stony Brook was placed 19th in the world and 10th in the United States.

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