Lecture Notes math104-s21
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2026-04-17T08:27:52+00:00Lecture Notes
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For HW7, we found a few students copying / paraphrasing the solution manual of Rudin. You can take a look [of the solution to Ch 4 here]. Let me be clear.
* First of all, this is cheating and won't be tolerated. The first time you will only get a warning, and second time you will get an F for this class.text/html2026-02-21T14:41:13+00:00Anonymous (anonymous@undisclosed.example.com)Extra Credit Assignment
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Extra Credit Assignment
If you want to start preparing for the final, and want to share your tips and troubles with each other, and
also want to get extra credit, you can participates in this 'Open Notes/Questions'.
The motivation is the following: if you did well in the midterms so far, then here is the chance that you can share your experience with others and share your understanding of this subject. If you did not do so well, and want to start preparing for the final and review the materia…text/html2026-02-21T14:41:13+00:00Anonymous (anonymous@undisclosed.example.com)math104-s21:final-stat
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96 A
92 A
92 A
91 A
90 A
89 A
89 A
86 A
84 A
84 A
82 A
78 B
77 B
77 B
76 B
74 B
73 B
72 B
70 B
69 B
67 B
65 B
63 B
62 B
59 B
58 B
58 B
57 B
56 B
55 B
53 B
52 B
51 B
49 B
49 B
46 C
46 C
44 C
44 C
44 C
42 C
42 C
42 C
41 C
40 C
37 C
37 C
31 D
27 D
11 F
6 Ftext/html2026-02-21T14:41:12+00:00Anonymous (anonymous@undisclosed.example.com)Final Exam
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Final Exam
Logistic
The final will be released on gradescope. You can begin the exam on gradescope anytime from May 12th, 9am to May 13th 12:00noon (hence, you need to submit the exam by 15:00 May 13th )
You will have 3 hours and 10 minutes to finish and upload your exam, starting from the moment that you open the exam. DSP students will automatically have extended alloted time (and the submission deadlines are extended accordingly)text/html2026-02-21T14:41:13+00:00Anonymous (anonymous@undisclosed.example.com)HW 1
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HW 1
* Read Ross section 3, and section 4 , especially density of $\mathbb{Q}$ in $\R$.
* Read Prof Hutchings' story about proof
* Ross: 1.1, 1.3, 1.6, 2.3, 4.7, 4.14, 4.15
* Challenge: 2.7 (just for fun, no need for submission)text/html2026-02-21T14:41:13+00:00Anonymous (anonymous@undisclosed.example.com)HW 2
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HW 2
* Read Ross section 7-9, especially p51-p54 where $\lim = \pm \infty$ are discussed.
* Do the exercises, but no submssions are required
* 7.1, 7.2, 7.3
* Submit the following problems
* 7.4, 7.5(a)
* 8.2（e), 8.4
* 9.1 ( c) , 9.2(b), 9.3, 9.4, 9.9( c)text/html2026-02-21T14:41:13+00:00Anonymous (anonymous@undisclosed.example.com)HW 3
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HW 3
In this week, we finished section 10 on monotone sequence and Cauchy sequence, and also touches a bit on constructing subsequences. It is important to understand the statements of the propositions/theorems that we talked about in class, and then try to prove them yourselves, then compare with notes and textbook.text/html2026-02-21T14:41:13+00:00Anonymous (anonymous@undisclosed.example.com)HW 4
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HW 4
* 11.5
* 11.10
* 11.11
* 12.10
* 12.11
* 12.12
* 12.13
* 12.14text/html2026-02-21T14:41:13+00:00Anonymous (anonymous@undisclosed.example.com)math104-s21:hw5-mistakes
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* 13.7
* (a) Only proving the statement in one direction
* (b) $E$ is not closed doesn't mean $E$ is open. There is no dichotomy between open and closed sets. Some sets maybe both open and closed, like $\emptyset$, some sets maybe neither open and closed, like $[0,1)$ in $\R$. And the notion of open and closed is a property of subset, so we need to say the full sentence like, 'the subset $A \In S$ is closed in $S$'. Say a set is closed or open without saying what's the ambient space has n…text/html2026-02-21T14:41:13+00:00Anonymous (anonymous@undisclosed.example.com)HW 5
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HW 5
Since we had a midterm this week, and did not cover much material, this homework is relatively short.
It will be based on first half of Ross section 13. [Rudin's chapter 2] also has a nice exposition, about metric space and topology.
* Ex 13.3, 13.4, 13.6, 13.7text/html2026-02-21T14:41:13+00:00Anonymous (anonymous@undisclosed.example.com)HW 6
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HW 6
We have two themes this week: the first one is the topology of metric space side. We discussed the notion of compact set. I recommend Rudin's book ch2. There are two major theorems, the Heine-Borel theorem, which applies to $\R^n$; and the compactness = sequential compactness, which is true for general metric space. See Rudin Theorem 2.41 (page 40)text/html2026-02-21T14:41:13+00:00Anonymous (anonymous@undisclosed.example.com)HW 7, P6
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HW 7, P6
Here are submissions that we suspect of copying each other or from some common sources. If you are one of such solution, and believe this is a mistake in tagging and did this problem independently, please explain your thought process and about this similarity.text/html2026-02-21T14:41:13+00:00Anonymous (anonymous@undisclosed.example.com)HW 7
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HW 7
On Tuesday, we discussed continuity of maps, the three equivalent definitions of continuity. Then on Thursday after we reviewed some topologies for metric space, we showed that continuous maps sends a compact set to compact set.
$\gdef\In\subset$text/html2026-02-21T14:41:12+00:00Anonymous (anonymous@undisclosed.example.com)HW 8
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HW 8
This week we learned about uniform continuity, and the automatic upgrade from continuous function to uniform continuous function when the domain is compact. We also learned about that continuous function sends connected set to connected set.text/html2026-02-21T14:41:13+00:00Anonymous (anonymous@undisclosed.example.com)HW 9
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HW 9
1.(2 points) Let $a_{nm}$ be a double sequence of real numbers, such that $\lim_{m \to \infty} a_{nm} = 1$ and $\lim_{n \to \infty} a_{nm} = 0$. Is it true that $\lim_{n \to \infty} a_{nn} $ exists and is in $[0, 1]$? If you think this is true, prove it. Otherwise find a counter example.text/html2026-02-21T14:41:12+00:00Anonymous (anonymous@undisclosed.example.com)HW 10: Derivatives
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HW 10: Derivatives
1. (2 pt) One corollary of the intermediate value theorem for derivative is the following (Rudin page 109): If $f$ is differentiable on $[a,b]$, then $f'$ cannot have any simple discontinuities on $[a,b]$. Give a proof of this statement.text/html2026-02-21T14:41:13+00:00Anonymous (anonymous@undisclosed.example.com)HW 11
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HW 11
1. Let $f: \R \to \R$ be a differentiable function. Assume that $f$ is convex, namely for any $x, y \in \R$ and any $t \in [0,1]$, we have
$$ tf(x) + (1-t) f(y) \geq f(t x + (1-t) y).$$
Prove that $f'(x)$ is monotonously increasing.
2. Let $f: (0, \infty) \to \R$ be a twice differentiable function. Suppose $f$'' is bounded, and $f(x) \to 0$ as $x \to \infty$. Show that $f'(x) \to 0$ as $x \to \infty$.text/html2026-02-21T14:41:13+00:00Anonymous (anonymous@undisclosed.example.com)HW 12
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HW 12
1. (2 point) Show that if $f$ is integrable on $[a,b]$, then for any sub-interval $[c,d] \subset [a,b]$, $f$ is integrable on $[c,d]$.
2. (2 point) If $f$ is a continuous non-negative function on $[a,b]$, and $\int_a^b f dx = 0$, then $f(x)=0$ for all $x \in [a,b]$.text/html2026-02-21T14:41:13+00:00Anonymous (anonymous@undisclosed.example.com)Midterm 1
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Midterm 1
Samples: Sample problems
Actual problems: [A], [B]
Solutions: [solution]
Stat: stat
If you are willing to put in effort and to improve, here are some suggetions about how to improve.text/html2026-02-21T14:41:13+00:00Anonymous (anonymous@undisclosed.example.com)Midterm 2
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Midterm 2
In the second part of the course, we first covered basic notions of topology: open / closed / compact subset, in the framework of metric space (Rudin Ch2). Then, we studied the notion of continuous maps between metric spaces(Rudin Ch4). We give three characterizations of continuous maps, using $\epsilon-\delta$ language, using convergent sequences, and the most general notion:text/html2026-02-21T14:41:13+00:00Anonymous (anonymous@undisclosed.example.com)Sample Problems
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Sample Problems
* A sample problem and solution from past course. We did not cover yet series, so you can skip problem 2 and 8.
* Here are some more practice problems (excerpt from other problem books) on sequences, [problems], [solution]. It is good to first try to discuss the problems in our zoom chat channel, before looking up the solutions.text/html2026-02-21T14:41:13+00:00Anonymous (anonymous@undisclosed.example.com)math104-s21:mt1-stat
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The overall grade in the end depends on your actual score, not the letter grade of the midterms. There are no A+ / A- etc assigned for midterms.
100 A
100 A
98 A
98 A
98 A
96 A
95 A
95 A
93 A
92 A
92 A
92 A
91 A
83 B
81 B
80 B
79 B
79 B
78 B
78 B
77 B
76 B
72 B
70 B
69 B
68 B
67 B
61 C
58 C
57 C
57 C
56 C
56 C
54 C
53 C
53 C
52 C
52 C
49 C
49 C
47 C
45 C
43 D
42 D
41 D
40 D
39 D
38 D
37 D
37 D
35 D
34 D
33 D
33 D
32 D
31 D
31 D
29 D
26 D
21 D
17 Dtext/html2026-02-21T14:41:12+00:00Anonymous (anonymous@undisclosed.example.com)math104-s21:mt2-stat
https://courses.pzhou.org/doku.php?id=math104-s21:mt2-stat&rev=1771684872&do=diff
Please check your zoom direct message for your graded exam files.
Average: 61.8
Median: 67
100 A+
96 A
95 A
92 A
92 A
90 A
89 A
88 A
87 A
83 A
80 A
80 A
77 B+
76 B+
75 B
74 B
74 B
73 B
73 B
73 B
71 B
70 B
69 B
68 B
66 B
65 B
63 B
63 B
62 B
60 B
59 B
58 B
58 B
56 B
55 B-
54 B-
52 C+
51 C+
49 C
49 C
49 C
44 C
42 C
40 C
40 C
40 C
39 C
39 C
37 C-
36 C-
25 D+
25 D
18 D
8 Dtext/html2026-02-21T14:41:13+00:00Anonymous (anonymous@undisclosed.example.com)Math 104: Introduction to Real Analysis (Spring 2021)
https://courses.pzhou.org/doku.php?id=math104-s21:start&rev=1771684873&do=diff
Math 104: Introduction to Real Analysis (Spring 2021)
Instructor: Peng Zhou
Email: pzhou.math@berkeley.edu
Welcome to Math 104, introduction to real analysis. We plan to cover the following topics:
"
"The real number system. Sequences, limits, and continuous functions in R and R. The concept of a metric space. Uniform convergence, interchange of limit operations. Infinite series. Mean value theorem and applications. The Riemann integral.text/html2026-02-21T14:41:13+00:00Anonymous (anonymous@undisclosed.example.com)suggetions about how to improve
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suggetions about how to improve
Hi everyone,
You may feel crushed after the first midterm. Indeed, it was too little time and problems are hard (or at least, no so familiar). The test was meant to be hard, and serves as a wake-up call. And here are some suggestions to improve.text/html2026-02-21T14:41:13+00:00Anonymous (anonymous@undisclosed.example.com)Math 104 and 113 Drop-In Tutoring Now Available via Zoom!
https://courses.pzhou.org/doku.php?id=math104-s21:tutor&rev=1771684873&do=diff
Math 104 and 113 Drop-In Tutoring Now Available via Zoom!
The Math Department is piloting a remote, drop-in tutoring program for students in Math 104 and 113. The department has hired well-qualified undergraduate tutors to support their peers in these courses based on the textbooks being used in sections. Details for tutor schedules and textbooks supported can be found at