UC Davis Math 125A
Real Analysis

Basic information

CRN codes: 40162 (Section A01) and 40163 (Section A02).
Room/Time of the Lectures: 09:00-09:50 a.m. MWF in Wellman 226
Discussion Section: 06:10-07:00 p.m. (section A01) and 07:10-08:00 p.m. (section A02) each Thursday in Olson 146
Instructor: Professor Alexander Soshnikov
Office: 3140 Mathematical Sciences Building
Office Hours: R 03:10-05:00 p.m.
E-mail: soshniko@math.ucdavis.edu
 
TA : Nam Lam
Office: 3127 Mathematical Sciences Building
Office Hour: W 01:30-02:30 p.m.
E-mail: namlam@math.ucdavis.edu
 
Midterm : Friday, February 17, in class.
Final: Friday, March 23, 01:00-03:00 p.m. in Giedt 1003.
Grading: Problem sets 25%, Midterm 30%, Final 45%
Webpage: http://www.math.ucdavis.edu/~soshniko/125a
 

Textbook:

Elementary Real Analysis (Second Edition, 2008) by Brian S. Thomson, Judith B. Bruckner, and Andrew M. Bruckner

pdf file of the textbook available at http://classicalrealanalysis.com/download.aspx

Other useful textbooks:

Elementary Analysis. The Theory of Calculus by Kenneth A. Ross and Understanding Analysis by Stephen Abbott

Syllabus

If you have a problem with the file, please download the syllabus from http://www.math.ucdavis.edu/courses/syllabi

From the departamental syllabi page:

ALERT: Effective Winter 2010, the Department is in discussion about the use of a different textbook than what is noted on the suggested syllabus for this course. As always with any class, please consult with your instructor for more information.

Homework

Homework will be assigned each Friday and will be due in class one week later.

No late submission allowed except for cases of true emergency.

While the students are allowed (and encouraged!) to discuss homework problems in groups, each student has to write down his/her own homework!

Extra problems (i.e. problems not from the textbook) are part of the hws and WILL contribute to the grade.

Homework 1 (due Friday, January 20):

Reading:

Section 5.1 (except subsection 5.1.3) and Section 5.2.

Problems from the textbook:

5.1.3, 5.1.10, 5.1.12 (pages 182-183), 5.1.16, 5.1.21 (page 185), 5.1.27, 5.1.29 (pages 187-188), 5.1.36, 5.1.37 (page 189), 5.2.10 (page 194)

Additional Problem:

Let g(x) be a real-valued function defined on a subset A of the real line R and let f(x) be a bounded function on the domain of g (i.e. there exists a positive number M such that |f(x)| < M for all x in A). Show that if g(x) has zero limit at x=c where c is a point of accumulation of A then f(x)g(x) also has zero limit at x=c.

Homework 2 (due Friday, January 27):

Reading:

Sections 5.4-5.6.

Problems from the textbook:

5.4.7, 5.4.13, 5.4.18, 5.4.20, 5.4.24, 5.4.30, 5.5.2, 5.5.4, 5.6.3.

Additional Problems:

none.

Homework 3 (due Friday, February 3):

Reading:

Sections 5.6-5.9

Problems from the textbook:

5.6.9, 5.6.11, 5.7.1, 5.7.2, 5.7.4, 5.7.6, 5.8.1, 5.8.6, 5.8.7, 5.9.1, 5.9.2, 5.9.8.

Additional Problems:

none.

Homework 4 (due Friday, February 10):

Reading:

Sections 7.1-7.5.

Problems from the textbook:

7.2.5, 7.2.7, 7.2.9, 7.2.18, 7.2.23, 7.3.7, 7.3.10, 7.3.14, 7.3.18, 7.3.19, 7.4.2.

Additional Problem:

Let f be a real-valued function defined on the real line. Prove that the set of all points of discontinuity of f can be represented as a countable union of closed sets.

Hint:

1) Let a>0. Call f a-continuous at c if there exists \delta>0 such that |f(x)-f(y)| < a for all x,y such that |x-c| < \delta, |y-c| < \delta.

2) Define D(a,f) as the set of all points on the real line where f is not a-continuous.

3) Prove that for any fixed a > 0 the set D(a,f) is closed.

4) Prove that if a < b then D(b,f) is a subset of D(a,f).

5) Let D(f) is the set of all points of discontinuity of f. Show that D(a,f) is a subset of D(f) for any a >0.

6) Prove that D(f) is the union of a countable family of sets D(1/n,f) where n ranges over all positive integers.

Homework 5 (due Friday, February 17):

Reading:

Sections 7.6, 7.7, 7.9, 7.10, 7.12.

Problems from the textbook:

7.5.2, 7.5.4, 7.6.1, 7.6.5, 7.6.9, 7.6.17, 7.9.1, 7.9.2, 7.9.3.

Additional Problem:

None. However, please do as many problems from the textbook as you can to prepare for the midterm!

Midterm (Friday, February 17)

will be based on the reading material of the first five homeworks.

Midterm Solutions

Midterm Grading Curve:

37-40 points: A+, 25-34 points: A, 22-23 points: A-, 20-21: B+, 18-19: B, 16-17: B-, 15: C+, 12-14: C, 11: C-, 10: D+, 9: D, 8: D-, 0-6: F.

Homework 6 (due Friday, February 24):

Reading:

Sections 7.10, 7.12, 9.1, 9.2.

Problems from the textbook:

7.10.6, 7.12.4, 7.13.2, 7.13.4, 7.13.7, 7.13.13, 9.2.1, 9.2.7.

Additional Problem:

none.

Homework 7 (due Friday, March 2):

Reading:

Sections 9.3, 9.4, 9.6

Problems from the textbook:

9.2.10, 9.3.1, 9.3.2, 9.3.5, 9.3.8, 9.3.11, 9.3.15, 9.3.18, 9.3.20, 9.4.1, 9.4.3, 9.4.10.

Additional Problem:

none.

Homework 8 (due Friday, March 9):

Reading:

Sections 9.6, 10.1, 10.2, 10.3.

Problems from the textbook:

9.6.1, 9.6.2, 10.2.1 c),d), 102.5, 10.2.6, 10.2.8, 10.2.12, 10.2.13, 10.3.1, 10.3.2, 10.3.3.

Additional Problem:

none.

Homework 9 (due Friday, March 16):

Reading:

10.4-10.6, 13.1-13.2.

Problems from the textbook:

10.4.3, 10.4.4, 10.4.5, 10.4.6, 10.5.3, 10.5.4, 10.5.6, 10.5.7, 10.6.1, 10.6.6.

Additional Problem:

Finish the proof of Theorem 10.10 (which means, pretty much, prove the result stated in Exercise 9.3.26 and apply it to Theorem 10.10).

Lectures:

Lecture 1 (January 9):

Introduction to Limits (section 5.1).

Lecture 2 (January 11):

Introduction to Limits (section 5.1).

Lecture 3 (January 13):

Properties of Limits (section 5.2).

Lecture 4 (January 18):

Continuity (section 5.4).

Lecture 5 (January 20):

Properties of Continuous Functions (section 5.5).

Lecture 6 (January 23):

Uniform Continuity (section 5.6).

Lecture 7 (January 25):

Extremal Properties (section 5.7).

Lecture 8 (January 27):

Darboux Property (section 5.8).

Lecture 9 (January 30):

Points of Discontinuity (section 5.9).

Lecture 10 (February 1):

How Many Points of Discontinuity? (section 5.9)

Lecture 11 (February 3):

Introduction (section 7.1). Derivatives (section 7.2). Continuity of the Derivative? (section 7.4).

Lecture 12 (February 6):

Computations of Derivatives. Algebraic Rules. The Chain Rule. Inverse Functions. (section 7.3).

Lecture 13 (February 8):

Local Extrema (section 7.5). Rolle's Theorem. Mean Value Theorem (section 7.6).

Lecture 14 (February 10):

Cauchy's Mean Value Theorem (section 7.6). The Darboux Property of the Derivative (section 7.9).

Lecture 15 (February 13):

Derivative of an Inverse Function (section 7.9). Convexity (section 7.10).

Lecture 16 (February 15):

Taylor's Theorem (section 7.12).

Midterm (Friday, February 17).

Lecture 17 (February 22):

Sequences and Series of Functions. Introduction (section 9.1). Pointwise Limits (section 9.2).

Lecture 18 (February 24):

Uniform Limits (section 9.3).

Lecture 19 (February 27):

Uniform Convergence and Continuity (section 9.4).

Lecture 20 (February 29):

Uniform Convergence and Derivatives (section 9.6).

Lecture 21 (March 2):

Introduction (section 10.1). Power series (section 10.2).

Lecture 22 (March 5):

Uniform Convergence (section 10.3).

Lecture 23 (March 7):

Functions Represented by Power Series (section 10.4).

Lecture 24 (March 9):

The Taylor Series (section 10.5).

Lecture 25 (March 12):

Analytic Functions (section 10.5). Products and Quotients of Power Series (section 10.6).

Lecture 26 (March 14):

Metric Spaces (sections 13.1 and 13.2).

Lecture 27 (March 16):

Convergence (section 13.4).

Lecture 28 (March 19):

Final Exam Preview.

FINAL EXAM: Giedt 1003, Friday, March 23, 01:00-03:00 p.m.