Math 125A: Real Analysis, Fall 2012

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Announcement: Final

Final scores and course grades are posted on Smartsite. Solutions to the final are here.

Important Dates

• Instruction begins: Thu, Sep 27
• Last day to add: Fri, Oct 12
• Last day to drop: Wed, Oct 24
• Last class: Fri, Dec 7
• Academic holidays: Mon, Nov 12; Thu-Fri, Nov 22-23

Exams

There will be two Midterms and a Final

• Midterm 1: Wednesday, Oct 24 (in class)
• Midterm 2: Wednesday, Nov 28 (in class)
• Final: Mon, Dec 10, 10:30 a.m.–12:30 p.m. (Exam Code B)

Grade

Grade will based on exams and homework, weighted

• 10%: Homework
• 25%: Each Midterm
• 40%: Final

Text

Text: Elementary Real Analysis, Thompson, Bruckner, and Bruckner, Second Edition, 2008.

The text is available for free download from Clasical Real Analysis.

Syllabus and Notes

A syllabus and list of topics for the class is here.

Some class notes are here:

• Contents. Contents
• Chapter 1. Properties of R
• Chapter 2. Limits of Functions
• Chapter 3. Continuous Functions
• Chapter 4. Differentiable Functions
• Chapter 5. Sequences and Series of Functions
• Chapter 6. Power Series
• Chapter 7. Metric Spaces

A complete updated set of notes is here.

Midterm 1

Solutions to Midterm 1 are here.

A detailed list of the topics for Midterm 1 is here. Some practice midterm questions are here. Solutions to the practice midterm questions are here.

Midterm 2

Solutions to Midterm 2 are here.

A detailed list of the topics for Midterm 2 is here. Some practice midterm problems are here. Solutions to the practice problems are here.

Final

Solutions to the Final are here.

A detailed list of the topics for the Final is here. Some practice problems are here. Solutions to the practice problems are here. As always, do the problems yourself before you look at the solutions.

Homework

Homework will be assigned weekly and a hard copy will be due in class on Friday. Please write clearly, or type, and staple your solutions.
If you want to type good-looking mathematics, the standard tool is LaTeX, or one of its many variants, and this would be as good a time as any to learn how to use it. See here to get started.
Problem numbers refer to the exercises in the text.

• Set 1 (Fri, Oct 5)
  Sec 5.1.1, p. 182: 5.1.1, 5.1.11
  Sec 5.1.2, p. 184: 5.1.16, 5.1.20
  Sec 5.1.4, p. 187: 5.1.28
  Sec 5.1.5, p. 189: 5.1.36
  Sec 5.2.3, p. 194: 5.2.10, 5.2.16
  Sec 5.2.4, p. 197: 5.2.17, 5.2.20
  Sec 5.2.5, p. 199: 5.2.27
  Read Sec 5.2 on "Properties of limits."

  Selected solutions on SmartSite are here.

  Set 2 (Fri, Oct 12)
  Sec 5.4.2, p. 212: 5.4.7, 5.4.13
  Sec 5.4.3, p. 214: 5.4.17, 5.4.20
  Sec 5.4.4, p. 217: 5.4.23, 5.4.24
  Sec 5.5, p. 218: 5.5.2, 5.5.4
  Sec 5.9.1, p. 225: 5.9.2, 5.9.6, 5.9.9
  Sec 5.9.2, p. 229: 5.9.11, 5.9.12
  Read Sec 5.9.1 on "Types of discontinuities" and Sec 5.9.2 on "Monotonic functions."

  Selected solutions on SmartSite are here.

  Set 3 (Fri, Oct 19)
  Sec 5.6, p. 221: 5.6.1, 5.6.3, 5.6.11
  Sec 5.7, p. 223: 5.7.1, 5.7.2, 5.7.4, 5.7.6
  Read the notes here that go over the definitions and properties of open, closed, and compact sets.

  Selected solutions on SmartSite are here.

  Set 4 (Fri, Oct 26)
  Sec 5.8, p. 224: 5.8.1, 5.8.2, 5.8.6, 5.8.9
  Sec 4.2.4, p.151: 4.2.5
  Sec 4.3.2, p.157: 4.3.5
  Sec 4.5.5, p.171: 4.5.2, 4.5.6

  Selected solutions on SmartSite are here.

  Set 5 (Fri, Nov 2)
  Sec 7.2, p.274: 7.2.3 (a),(d), 7.2.5, 7.2.12, 7.2.15
  Sec 7.2.2, p.276: 7.2.16
  Sec 7.2.3, p.278: 7.2.26
  Sec 7.3.1, p.280: 7.3.1, 7.3.4
  Sec 7.3.2, p.284: 7.3.15
  Sec 7.3.3, p.286: 7.3.18

  Selected solutions on SmartSite are here.

  Set 6 (Fri, Nov 9)
  Sec 7.5, p.291: 7.5.1, 7.5.2, 7.5.4
  Sec 7.6.1, p.293: 7.6.1, 7.6.2, 7.6.4
  (There's a typo in in 7.6.2: the equation should be x^3 + ax + b = 0.)
  Sec 7.6.2, p.295: 7.6.8, 7.6.11, 7.6.13
  Sec 7.7, p.299: 7.7.3, 7.7.4

  Selected solutions on SmartSite are here.

  Set 7 (Fri, Nov 16)
  Sec 7.12, p.322: 7.12.1, 7.12.5, 7.12.6 (You can assume l'Hospital's rule, Th. 7.42)
  Sec 9.2, p.371: 9.2.1, 9.2.4, 9.2.7
  Sec 9.3, p. 381: 9.3.1, 9.3.2, 9.3.3

  Selected solutions on SmartSite are here.

  Set 8 (Fri, Nov 30)
  Sec 9.3, p. 381 9.3.11, 9.3.13, 9.3.18, 9.3.20, 9.3.22
  Sec 9.4, p. 386 9.4.1, 9.4.3, 9.4.10
  Sec 9.6, p. 396 9.6.1, 9.6.2
  Review material on series (Chapter 3 of the text), especially: Cauchy criterion; absolute convergence; comparison test; ratio test.

  Set 9 (Fri, Dec 7. Will not be collected.)
  Sec 10.2, p. 408: 10.2.1
  Sec 10.3 p. 411: 10.3.1
  Sec 10.4 p. 417: 10.4.5
  Sec 10.5 p. 423: 10.5.4
  Sec 13.2 p. 541: 13.2.2, 13.2.10
  Sec 13.4 p. 548: 13.4.7, 13.4.14 (a), (b), (c)
  Sec 13.5 p.555: 13.5.9, 13.5.10
  Sec 13.6 p. 562: 13.6.16

Some Extra Material

Plots of Some Functions. The function y = sin(1/x) and a detail near the origin. The function y = x sin(1/x) and a detail near the origin. The Thomae function, equal to zero for irrational numbers and 1/q for rational numbers p/q with p, q relatively prime, and a blow-up for small y-values

Definition of Continuity. If you find the ε-δ definition of continuity a bit unintuitive, you might like this article by Keith Devlin.