# Welcome to Math 4130!

Our textbook is R. Strichartz, The Way of Analysis, revised edition (2000).

Syllabus is available here.

# Quizzes

Quiz 1, 20 minutes: Theory of limits, open and closed sets. March, 2. The quiz costs 1 homework.

Quiz 2, 20 minutes: Limits of functions, continuous and monotonic functions, derivatives. The quiz costs 1 homework.

Quiz 3, 20 minutes: Riemann integrals (see Summary 6.4). The quiz costs 1 homework.

# Home assignments:

Supplementary problems are optional and give extra credit.

HW1, due February, 2.

HW2, due February, 9.

HW3, due February, 16.

HW4, due February, 23.

HW5, due March, 2. Solutions to HW5

HW6, due March, 14.

Supplementary homework on metric spaces, due March, 16.

HW7, due March, 23.

HW8, due March, 30; you may use familiar formulas for derivatives of sine and cosine.

HW9, due April, 13.

HW10, due April, 20.

HW11, due April, 27.

HW12, due May, 4.

HW13, due May, 18 (the beginning of the final).

You may submit HW13 earlier if you need a feedback on your solutions before the final.

# Final: May, 18, 2:00 PM – 4:30 PM, MLT203. Solutions to Final

Closed-book final. You may use a letter-size paper with your notes.

Lecture 1: Quantifiers and sets (Sections 1.1, 1.2.1). Optional reading: Section 1.3 (Proofs).

Lecture 2: Uncountable sets (Sec 1.2.2). Rationals as an ordered field (Sec 1.4). Optional reading: Section 1.5 (Axiom of choice).

Lecture 3: Real numbers as decimals. Cauchy sequences of rationals and the definition of real numbers (Sec 2.1)

Lecture 4: R as an ordered field (Sec 2.2)

Lecture 5: Limits. Completeness of R (Sec 2.3)

Lecture 6: Square roots (Sec 2.3.2). Limits and limit points. (Sec 3.1)

Lecture 7: Supremums and infinums (Sec 3.1). Closed and open sets (Sec 3.2)

Lecture 8: Open, closed, and compact sets (Sec 3.2, Sec 3.3 -- first two definitions of compact sets).

Lecture 9: Limits of functions. Continuous functions. Uniform continuity. (Sec 4.1).

Lecture 10: Limits of functions. Continuous functions. (Sec 4.1).

Lecture 11: Properties of continuous functions (Sec 4.2)

Lecture 12: Properties of continuous and monotonic functions (Sec 4.2). The third definition of the compact set (Sec 3.3). The definition of 2^x.

Lecture 13: Derivative and differentiability (Sec 5.1). Properties of the derivative, Mean value theorem (Sec 5.2).

Lecture 14: Intermediate value theorem (Sec 5.2.2). Calculus of derivatives, inverse function (Sec 5.3).

Lecture 15: Higher derivatives. Taylor expansions (Sec 5.4)

Lecture 16: Integrals of continuous functions (Sec 6.1)

Lecture 17: Integrals of continuous functions (Sec 6.1). Riemann integral (Sec 6.2).

Correction to the lecture 17: for the last theorem, the sketch of the proof I gave only works if the set of I_n is finite; see the next lecture.

Lecture 18: Lebesgue theorem about Riemann integrability (close to Sec 6.2.3). Improper integrals (Sec. 6.3). Complex numbers (Sec 7.1).

Lecture 19: Complex numbers. Complex-valued functions (Sec 7.1). Numerical series (Sec 7.2).

Lecture 20: Rearrangements of absolutely and relatively convergent series (Sec 7.2). Uniform convergence (Sec 7.3).

Lecture 21: Integrals and derivatives for uniformly convergent sequences (Sec 7.3). Power series (Sec 7.4).

Lecture 22: Power series and analytic functions (Sec 7.4). Exponential and logarithm (Sec 8.1).

Lecture 23: Exponential and logarithm (Sec 8.1). Trigonometric functions (Sec 8.2)

Lecture 24: Equicontinuity and Arzela-Ascoli theorem (Sec 7.6)