Math 104 - Introduction to Analysis

InstructorPaul Vojta
LecturesMWF 2–3, Cory 247
Class Number40152
Office883 Evans
E-Mailvojta@math.berkeley.edu
Office HoursMondays 12:30–1:30, Wednesdays 1–2, Fridays 1–2,
excluding spring break and university holidays.
Please email me to set up a time to meet if you cannot make any of these times.
GSI Office HoursThe GSI for Math 104 (all sections) is Ben Wormleighton. He will also hold office hours this semester for Math 104 students. His office hours will be:
Monday1–3 Evans 869
4–6Evans 891
Thursday9–12 Evans 891
Friday9–12 Evans 891
PrerequisitesMath 53 and 54
Required Text Jiří Lebl, Basic Analysis: Introduction to Real Analysis. See below for availability.
Catalog DescriptionThe real number system. Sequences, limits, and continuous functions in R and Rn. The concept of a metric space. Uniform convergence, interchange of limit operations. Infinite series. Mean value theorem and applications. The Riemann integral.
Syllabus TBD
GradingGrading will be based on:
30%Homework Assigned weeklyDue in class, usually on Wednesdays
15%First midterm February 282:00–3:00 pm
20%Second midterm April 42:00–3:00 pm
35%Final examTuesday, May 8 11:30 am–2:30 pm
HomeworkAssigned weekly, generally due on Wednesdays. Assignments are given below. Solutions will be posted on bCourses.
CommentsI tend to follow the book rather closely, but try to give interesting exercises and examples.
This course is a bit harder than some of the other basic upper-division math courses, such as Math 110 or perhaps 113. I recommend against taking this course if you haven't had some other upper-division math course or Math 55.
Note the final exam date given above (exam group 6). Do not enroll in this course if you cannot take the exam at that date and time, whether because of a conflict, too many exams on that day, or any other reason. The schedule of final exams is available at http://registrar.berkeley.edu/sis-SC-message.

The Textbook

The textbook is Basic Analysis: Introduction to Real Analysis, by Jiří Lebl, and is available for free on the web. For this class we will use the edition of February 29, 2016 (version 4.0). You can get it either:

Note that a list of errata for the book is available.

Homework Assignments

General rules on homework assignments are:

Homework assignments are due in class on the days indicated below.

No.DueDownload Comments
1January 24 pdf dvi
2January 31 pdf dvi
3February 7 pdf dvi
4February 14 pdf dvi
5February 21 pdf dvi
6March 2 pdf dvi
7March 7 pdf dvi
8March 14 pdf dvi
9March 21 pdf dvi
10April 6 pdf dvi Correction: In the first problem, you also need to assume that φ is monotone increasing.
11April 11 pdf dvi Problem 1 is Exercise 7.2.9 in the textbook. You will receive no credit for merely pointing this out.
12April 18 pdf dvi Correction for Exercise 7.5.10: at the end it should say inf{d(x,y):  xK1, yK2}.
13April 25 pdf dvi
14May 2
Do not hand in
pdf dvi Solutions are now posted on bCourses

Course Handouts

No.DateTitle Download
1January 17 Equivalence of three variations on induction pdf dvi
2February 2 Infinite limits pdf dvi
3March 7 Withdrawn Nested intervals theorem and Heine-Borel theorem pdf dvi
4March 12 An increasing function discontinuous at all rational numbers pdf dvi
5March 14 The Heine–Borel theorem for [a,b] pdf dvi
6March 21 Cauchy–Schwarz and the triangle inequality pdf dvi
7March 23 Overview of limits and the extended reals pdf dvi
8April 20 Open subsets of R pdf dvi
9April 23 Supplemental material on power series pdf dvi

Exams (Generally)

Policies for exams are as follows.

The two midterms will be given during the normal class hours (2–3 pm), and will be in our normal classroom (Cory 247).

Generally, about a week before each exam, a sample exam will be distributed in class and posted on bCourses. This will usually be an exam from an earlier Math 104 class that I've taught. Sample exams should be used to get a general idea of the likely length of an exam and the general nature of questions to be asked (e.g., the balance between computational and more theoretical questions). However, one should not (for example) note that a sample exam contains questions on material from Sections 1.5, 2.1, 2.7, 3.1, 3.4, etc., and expect to see questions from those sections on the actual exam.

Exams are cumulative, so the second midterm may have questions from material prior to the first midterm. Of course, the final exam will cover the whole course, but will have increased emphasis on the material not covered on the midterms.

Here is a link "How to lose marks on math exams" (by a former GSI Andrew Critch).

The Math Department maintains an archive of old exams (usually without answers). Here is the link for Math 104.

And finally, a word about regrades: Grade calculation errors are welcome for discussion or review. Whether this solution should be worth 4 or 5 points is not.

First Midterm

The first midterm was given on Wednesday, February 28, from 2:10 to 3:00, in our usual classroom. It covered everything in the textbook up to and including Example 2.5.4 (page 77), plus added topics from the lecture (notably, infinite limits, infinite lim sups, etc.), minus Section 1.5 (decimal representations of real numbers). It also covered material in lectures and handouts up to and including February 16, and homework assignments up to and including Homework #5.

A sample midterm, solutions for it, and solutions for the actual midterm are available on bCourses.

The (very rough) curve for the midterm is:

A67–75
B45–66
C35–44

The median was 52, the mean was 51.4, and the standard deviation was 15.9.

Second Midterm

The second midterm was held on Wednesday, April 4, from 2:10 to 3:00, in our usual classroom. It covered everything in the textbook up to and including Section 3.4, plus added topics from the lecture (notably, infinite limits, infinite lim sups, Heine-Borel for closed intervals, etc.), minus Section 1.5 (decimal representations of real numbers) and parts of Section 2.6 (rearrangements of series, multiplication of series, and power series). It also covered material in lectures and handouts up to and including March 16, and homework assignments up to and including Homework #9.

A sample midterm, solutions for it, and solutions for the actual midterm are available on bCourses.

The (very rough) curve for the midterm is:

A74–100
B55–73
C36–54

The median was 61, the mean was 58.6, and the standard deviation was 19.75.

Final Exam

The final exam was held on Tuesday, May 8, from 11:30 (sharp) to 2:30, in our usual classroom (Cory 247). It covered all sections in the textbook as listed above for the second midterm, plus Sections 3.5, 7.1–7.5, 4.1–4.2, 5.1–5.3, and 6.1–6.2. (Note that the sections are listed in the order covered, not numerical order, so it does include the sections on metric spaces.) The final exam (potentially) covered all homework assignments, and all of the handouts with the following exceptions:

A sample exam, and solutions for it, are available on bCourses.

Homework Score

The overall homework score has now been posted on bCourses.

Note that the homework score has not been computed on a percentage basis. Homework scores were first adjusted so that the mean on each homework (among those turned in) was the same across homeworks. Then the top two were dropped. The resulting scores were added. Scores in roughly the bottom half of the class were brought up, so that homework would not have a disproportionate effect on the final grade. And, finally, the resulting score was adjusted to have a mean and standard deviation proportionate to the means and standard deviations of the midterms and finals (as if it were a hypothetical exam worth 150 points). Note, however that when bCourses says that the score is "out of 150", this does not mean that the highest homework score was 150, or even that someone getting perfect scores on all of their homeworks would get 150 points. It's just that bCourses requires a number there, so I put 150.


Last updated 14 May 2018