This course is a natural continuation of our Analysis I course. A rigorous understanding of sequence limits and continuity enables us to introduce other key concepts of Real Analysis: differentiation, integration and power series.
We have structured the course to provide you with opportunities to collaboratively prove all theorems with your peers. Several days before each class, you will receive lecture notes containing definitions, examples, and propositions that we need to prove. In class, we will discuss your approaches, comments, and any questions you may have.
Prerequisites
Concepts studied in the Foundations of Analysis and Analysis I courses.
Curriculum
- Definition of the derivative.
- Newton’s approximation.
- The Leibniz rule.
- Local maxima, local minima, and derivatives. Rolle’s theorem.
- Mean value theorem.
- Chain rule.
- Inverse functions and their derivatives.
- The exponential and logarithmic functions.
- L’Hôpital’s rule.
- Derivatives of higher order.
- Taylor’s theorem.
- Power series. Abel’s test.
- Differentiation of vector-valued functions.
- Partitions of a closed interval and piecewise constant functions.
- Upper and lower Riemann integrals.
- Basic properties of the Riemann integral.
- Riemann integrability of continuous functions.
- Riemann integrability of monotone functions.
- A non-Riemann integrable function.
- The Riemann-Stieltjes integral.
- The two fundamental theorems of calculus.
- Consequences of the fundamental theorems.
Grading & Assessment
There will be two classes each week: one lecture and one problem-solving class. The lecture is not a traditional presentation; it is a guided discussion of the pre‑read notes where we jointly prove the day’s theorems. We will use a collaborative online whiteboard (Miro) for each session where lecture notes or problem sets are uploaded in advance. Each week you should write solutions to at least four problems total across those boards. You will receive 1 point for each correct solution, up to a maximum of 4 points per week. Additional bonus points may be awarded for the shortest and most elegant solutions or for solutions to the most challenging exercises.
Both the midterm and final exams are oral, open-book, and open-notes. The midterm consists of 5 problems worth 3 points each, and the final exam consists of 5 problems worth 4 points each. You also receive 1 bonus point each time you are the first to correct the professor’s error during a lecture.
Grade = (regular points earned + bonus points) / (regular maximum possible points). Thus it is possible to score above 100%. To receive an A+, you need to earn at least 90% of regular points; an A requires 85%, and so on.
Textbooks
- T. Tao, Analysis I, 4th edition, 2022.
- S. Abbott, Understanding Analysis, 2nd edition, 2016.
- W. Rudin, Principles of Mathematical Analysis, 3rd edition, 1976.
- V. Zorich, Mathematical Analysis I, 2nd edition, 2015.
We recommend several textbooks below. Please select those that best meet your needs. You are not required to use the most recent editions; these are listed for your reference only.
Teasers and Spoilers
- Does there exist an everywhere differentiable function \( f:\mathbb{R}\to\mathbb{R} \) such that its derivative is discontinuous at some point?
- Does there exist a smooth function \( f:\mathbb{R}\to\mathbb{R} \) such that
\[ f^{(n)}(0)=0,\quad\forall n\ge 0, \]
and such that \( f \) is not a constant function? - Show that
\[ \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} = \ln 2. \] - Does there exist a Riemann integrable function with an infinite number of discontinuities?