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 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.