This is a publicly available information page about this course.
Registered students can find complete information in the course Canvas site.
If you are registered for the course and do not have access to this site by August 25, please contact me at tyson@illinois.edu.
Added May 7, 2026: Access to course lecture notes and final exam information.
Course Information
Lecture Times and Location
MWF 10-10:50, 143 Henry Administration Building
Course Instructor
Professor Jeremy Tyson
tyson@illinois.edu
104B Harker Hall
Textbook
There is no required textbook for the course. I will post my own written set of lecture notes to this Canvas site throughout the semester.
Here are several suggested (optional) texts.
- Introduction to Analysis by M. Rosenlicht (Dover, 1968)
- Real Analysis with Real Applications by K. R. Davidson and A. P. Donsig (Prentice-Hall, 2002)
One of these books (Rosenlicht) is available for purchase in the bookstore. The books by Rosenlicht and Davison-Donsig are both on reserve. Please note that Math Library course reserve books can be accessed at the Grainger Engineering Library Information Desk.
- Analysis I (Springer, 2022) and Analysis II (Springer, 2023), by Terence Tao. These two books are available as e-books through the University Library. The relevant chapters in these two books corresponding to what we cover in this class are Analysis I (Chapters 4-11) and Analysis II (Chapters 1-4).
- Advanced Calculus, by Piotr Hajlasz. Available at https://sites.pitt.edu/~hajlasz/Teaching/Math1530Fall2018/selection.pdf. Chapters 6 through 19 in these lecture notes correspond to what we cover in this class.
If you are interested in learning more about the history of calculus and the development of real analysis, I recommend the following two books:
A History of Analysis, edited by Hans Niels Jahnke, History of Mathematics volume 24, published by the American Mathematical Society and the London Mathematical Society, 2003.
The Calculus: A Genetic Approach, by Otto Toeplitz (University of Chicago Press, 1963, republished 2007)
Course Description
Math 424 is a rigorous treatment of real analysis. A major part of the course consists of the derivation of the basic results of first semester calculus (continuity, differentiation, and Riemann integration). The first half of the course develops a significant part of real analysis in the abstract setting of metric spaces. Topics which we will discuss include continuity, compactness, and connectedness. In the second half of the course we recapitulate calculus of a single real variable using the machinery of metric space analysis. We focus particularly on the interchange of limit operations.
This course is part of the Mathematics Honors Sequence. Eligibility for this section requires the approval of the Director of Undergraduate Programs in Mathematics.
Schedule of topics to be covered
| Set theory and cardinality; axiomatics and construction of the real numbers | 4 lectures |
|---|---|
| Metric spaces, topology, sequences, convergence, completeness, compactness, connectedness | 10 lectures |
| Limits and continuity, sequences of functions, completion of a metric space | 7 lectures |
| Differentiation, Mean Value Theorem, Taylor approximations | 4 lectures |
| Riemann integration, Fundamental Theorem of Calculus, special functions, characterization of Riemann integrable functions, sets of measure zero | 6 lectures |
| Interchange of limit operations, Leibniz theorem (differentiation under the integral sign), power series, convergence of power series | 4 lectures |
| Weierstrass’ continuous and nowhere differentiable function, functions of bounded variation and monotone functions, differentiability | 2.5 lectures |
| Banach Fixed Point Theorem, Picard’s theorem, iterated function systems | 2.5 lectures |
We may take more or less time on various topics as circumstances dictate.
Grades
The final grade will be computed according to the following percentages:
- Weekly Homework: 20% (lowest score will be dropped)
- Quizzes (approximately 1-2 per month): 10% (lowest score will be dropped)
- Midterm exams 1 and 2: 20% each
- Final exam (date determined by the University): 30%
Final averages will not be curved down. For example, if your final average is 80% then your final grade is guaranteed to be some sort of “B”.
It is possible that final averages may be curved up. Information about grade distribution will be provided during the semester, and the precise function which assigns a letter grade to each percentage will be determined at the end of the course.
Academic Integrity
Students are expected to adhere to the policies of the University regarding academic integrity as described in https://studentcode/illinois.edu/article1/part4.
You are encouraged to work on the homework in groups, although each student must write and submit their own separate set of solutions. Please indicate on your submitted homework which other students you collaborated with, if any.
In no case should solutions be copied from other sources (e.g. from the internet or from other students).
All work on exams must be done independently.
Resources for students with disabilities
To obtain disability-related academic adjustments or auxiliary aids, students with disabilities should contact the course instructor and Disability Resources and Educational Services (DRES). To contact DRES, please visit 1207 S. Oak St., Champaign, call 333-4604, e-mail disability@illinois.edu or visit the DRES website.