Math 104: Introduction to Analysis, Spring 2025

Instructor: Zhiyan Ding (zding.m at berkeley dot edu). In all e-mail correspondence, please include "[Math104]" in the subject line.

GSI: TBD

Lectures: LEC001: TuTh 9:30am-10:59am in Dwinelle 223

LEC002: TuTh 11:00am-12:29pm in Dwinelle 215

Text:

Elementary Analysis: The Theory of Calculus, by Kenneth A. Ross.

Students should feel free to consult other books for additional exercises and/or alternative presentations of the material

Students are expected to read the relevant sections of the textbook, as the lectures are meant to complement the textbook, not replace it, and we have a lot of material to cover.

Grading:

Choice 1: 20% homework, 2 x 20% in-class midterms, 40% final exam.

Choice 2: 20% homework, 20% in-class midterm (higher one), 60% final exam.

All exams are conducted in class and are closed book. For the two midterms, a single-page cheat sheet is allowed. For the final exam, a two-page cheat sheet is permitted.

There will be 12 homework. The lowest two homework scores will be dropped.

No makeups for the midterms will be given except in cases requiring special accommodation.

Course policies:

• Homework will be assigned regularly (see the syllabus) and due at the beginning of class. I grant extensions at most twice in reasonable circumstances, but you must talk to me as early as possible. The longer you wait, the less flexible I will be.

• The homework solution will be posted one week after the due date. Even if you request an extension, the maximum extension time allowed is 7 days.

• Collaborating or using AI on homework: You may work together or use AI to figure out homework problems, but you must write up your solutions in your own words in order to receive credit. In particular, please do not copy answers from the internet or solution manuals. Also, please do not let others copy your homework. You might lose points in homework if I find out your homework is the same as others.

The major purpose of the homework is to help you check your understanding of the material and prepare for the exams. I will try to make the difficulty of homework be similar to the exams. I highly encourage you to take some time to think about each problem by yourself (say, at least thirty minutes) before discussing it with others. If you're stuck on a problem, another choice is to come to office hours and ask for a hint. The more you figure out on your own, the better you'll do both on the exams.

• The usual expectations and procedures for academic integrity at UC Berkeley apply. Cheating on an exam will result in a failing grade and will be reported to the University Office of Student Conduction. Please don't put me through this.
  
  

Special announcements:

• If you are a disabled student (with or without a document from the Disabled Students' Program) and require special accommodations of any kind, please e-mail the instructor as soon as possible, and no later than the end of the second week of classes. It really helps if you tell me earlier rather than later.

• If you are representing the university on some official duties (say if you are an athlete or in a band), and if there is a conflict with any of the mid-terms, please let the instructor know before the end of the second week of classes.
  
  

Additional resources:

• Principles of Mathematical Analysis, by Walter Rudin

• Real Mathematical Analysis, by Charles Pugh

• Basic Analysis: Introduction to Real Analysis , by Jiří Lebl
  
  

Course Overview:

This is an introductory analysis course. In this class, we will start with the real number system. After that, we will revisit some classical concepts in calculus (e.g. limits, continuity, series, differentiation, Riemann integration, etc) and study them in a more rigorous way.

This is a proof-oriented math course. Some proof-writing experiences are helpful, but not necessary.

Syllabus:

The following topics will be covered in class.

• Real numbers: properties of the real number system, including the completeness axiom and the least upper bound property.

• Limits: definitions and properties of limits, including squeeze theorem and continuous functions.

• Derivatives: definition and properties of derivatives, including the mean value theorem and applications to optimization.

• Integrals: definition and properties of definite and indefinite integrals, including the fundamental theorem of calculus.

• Series: definition and convergence tests for series, including the ratio test and the root test.

• Sequences: definitions and convergence tests for sequences, including the Bolzano-Weierstrass theorem and the monotone convergence theorem.

• Metric spaces: definitions and examples of metric spaces, including convergence and completeness.