MATHEMATICS 341- SPRING 2013-COURSE LOG

W 1/9  Discussion of syllabus/ Limit of sequences: definitions, examples (2.2)
HW1: 2.2.2, 2.2.4

F   1/11 Limit of sequences: algebraic theorems (2.3)
HW1: 2.3.5, 2.3.6, 2.3.8, 2.3.11

M 1/14 sequences: comparison theorem, squeeze principle (2.3)
Ex: 2.4.2, 2.4.5, 2.2.1
HW1: 2.4.3, 2.4.4

W 1/16 sequences: monotone convergence (2.4), problems 2.4.2, 2.2.1
HW2: 2.5.3, 2.5.4, 2.5.5

F  1/18 Sequences: Prob 2.3.11/ lim sup/ lim inf of bounded sequences (2.4.6.(a),(b),(c))/Bolzano-Weierstrass theorem

Challenge problem: (4HW points): Let (a_n) be a bounded sequence. Show (i) If a subsequence of a_n converges, with
limit L, then L is less than or equal to lim sup a_n; (ii) There exists a subsequence of a_n converging to lim sup a_n.

HW2: 2.4.6(d)

M 1/21 MLK holiday (no classes)

W 1/23 sequences: Cauchy sequences, completeness
HW2: 2.6.1

F 1/25 UTK closed due to ice storm (HW2 due Monday)

M 1/28 Series: Cauchy criterion, divergence criterion, comparison theorem (positive terms),
absolute vs. conditional convergence, geometric series
HW3: 2.7.5, 2.7.6, 2.7.10

W 1/30 Series convergence tests: alternating series (2.7.1), polynomials (2.7.7), ratio test (2.7.9)

F  2/1: Series convergence tests: summation by parts,  Dirichlet's test, Abel's test

M 2/4: The Cantor set (3.1)/ zero length, uncountability, no intervals.

W 2/6: open sets, union/intersection of arbitrary families, accumulation points
HW4 (due 2/8) 3.2.2, 3.2.3 (you will need the def. of closed set, def 3.2.7), 3.2.7 and the problem:
Write a complete proof of the fact that there exists a bijection from the Cantor set to the set of all infinite sequences of 0s and 1s.

F 2/8: sln of problem above/ closed sets/ general unions and intersections/ interior points
HW5(due 2/15): 3.2.9, 3.2.12, 3.2.13, 3.4.1

M 2/11: interior and closure of a set (two constructions)/ perfect sets
HW5: (due 2/15): 3.3.4, 3.3.7 and the problems:
Problem A: Let A be a non-empty subset of the real line. Show that its interior int(A) is an open set.
Problem B: Let A be a non-empty subset of the real line. Show that int(A) is empty if, and only if, A contains no intervals.
Challenge problem (optional, 3pts.):
The interior int(A) of a non-empty set of the reals is "the largest" open set contained in A.
Define what this means precisely (three conditions) and give an "abstract" construction of int(A) (analogous to the abstract
construction of the closure of a set, given in class.)

W 2/13: compact sets/ perfect sets are uncountable

F 2/15: review; HW5 due

M 2/18 TEST 1 (Chapters 2 and 3)
test 1
test 1 solutions

W 2/20: discussion of test 1

F 2/22: limits at a point (4.2), continuity (4.3, start).
image and preimage of a set under a function/ removable, jump and oscillatory discontinuities (examples).
connected sets.
The general question in this chapter: if f:R--> R is continuous and a subset A of R is open (or closed, bounded, compact, connected)
is the image (or the preimage) of A under f  open (resp. closed, bounded, compact, connected)?
HW6 (due 3/1): 4.2.6, 4.2.8, 4.3.7, 4.3.11, 4.3.12

M 2/25:  continuous functions: examples (see question above).  Extreme Value Theorem.

W 2/27: continuity of composition/ intermediate value theorem/ monotone equivalent to invertible/ continuity of inverse function
HW6: 4.4.6, 4.5.7  (Hint for 4.5.7: if neither 0 nor 1 are fixed points, argue that g(x)=f(x)-x is positive at 0 and negative at 1.)

F 3/1: uniform continuity/ Lipschitz functions (Ex. 4.4.9)
Three problems for Monday, 3/4:
1. Let  F be a closed subset of R, with empty interior. Prove that the discontinuity set of the characteristic function of F is exactly F.
2. Let f: R--> R be continuous and strictly monotone (increasing or decreasing). Then if I is an open interval, f(I) is also an open interval.
3. Problem 4.3.9 (b)(c)(d) (contractions have fixed points). (No need to do part (a).)

M 3/4: continuous extension (Ex. 4.4.13)/ Discontinuities of monotone functions (4.6).

W: 3/6 review handout on continuous functions (examples and proofs: due Monday 3/11). Student survey (voluntary, but
must be returned by Friday 3/8).
Also due Monday 3/11 (HW 7): 4.6.4, 4.6.5 (outline given in class)
Discussed: problem 4.3.9 (contractions have unique fixed points)

F 3/8: 5.2: derivatives: definition, calculus
HW 8 (due Friday 3/15): 5.2.4, 5.2.5

M 3/11: Derivatives at endpoints of an interval; Darboux's theorem; discussion of problems 1 and 2 from 3/1

W 3/13: Mean value theorem and applications.
Discussion of 5.2.8 (Darboux's theorem): (a) T (b) F (T if f' continuous) (c) T (d) F (T if f continuous)
HW 8: 5.3.1, 5.3.4, 5.3.5, 5.3.7, 5.3.8

F 3/15: Review problems on the derivative (with solutions!)
(extension of problem 5.2.8 in the text. Correspondence with problems in text has been added. Due Monday 3/18)
Mapping properties of continuous functions (with solutions!)
(This is an improved version of the handwritten handout included in HW7, and solved in class on Friday.
It includes two new problems (optional). These will be discussed on Monday, and can be turned in in writing.)

Exercises 5.3.1 and 5.3.3 were discussed in class. Can you answer 5.3.3. (c)?

How to study for the test.  (i) Read the sections in the text including theorems/definitions discussed in class. (ii) Go for a walk, and organize these
results and definitions in your mind. (iii) Solve as many problems as possible (independently, without looking at solutions), particularly
problems from the text discussed in class or in the HW, and problems in these two online handouts.

M 3/18: review class
Keywords and results for this part of the course.

Intermediate Value Theorem, Extreme Value Theorem, topological properties of f(A) and f^{-1}(B)
Lipschitz condition, contractions, fixed points
Uniform continuity,  extension result, discontinuities of monotone functions
Darboux's theorem  and discontinuities of the derivative
Local max/min at interior points and at endpoints
Mean Value Theorem, L'Hospital's rule for indeterminate limits
Behavior of f in a neighborhood of c when f'(c) is not zero (handout problems)

W 3/20: Exam 2 (Chapters 4 and 5)
Exam 2 solutions

F   3/22 to F 3/29: SPRING BREAK

M 4/1: discussion of exam 2/ Pointwise vs. uniform convergence (pointwise limits need not be continuous).

W  4/3: uniform convergence: examples, uniform convergence on compact sets, continuity, Cauchy criterion
HW9 (due 4/5, Friday): 6.2.1 to 6.2.5 (5 problems)

F 4/5: discussion of HW problems 6.2.4, 6.2.5/ statement of monotone convergence and Ascoli-Arzela, examples.

M 4/8: Monotone convergence (6.2.12), equicontinuity (6.2.15), Ascoli-Arzela (6.2.16)

Remark: beginning on 4/3, (almost) every class will start with a short optional quiz, dealing with material discussed in class
recently (one or two questions, points count towards the HW grade)

Addendum to syllabus: the three highest grades (of the three exams and HW) will enter the course average with
weights 30, 20, 20. The final will have weight 30.

W  4/10: differentiability of the limit/ main theorem and extension (6.3.5).
HW10 (due 4/12, Friday): 6.3.1 to 6.3.4 (a)(b) (no need to do (c)) (4 problems)

F 4/12 6.4 (series of functions) Examples: 6.4.4, 6.4.5
HW 11 (due W 4/17) 6.4.3, 6.4.5, 6.4.7

M 4/15 6.5 (power series), 6.6 (Taylor series)
HW 11 (due W 4/17): 6.5.1
Bonus problems: 6.5.2, 6.6.5

W 4/17 Review session (also: F 4/19 at 3:30, location TBA)
Problems discussed:  6.2.2, 6.2.3, 6.2.4, 6.2.8, 6.3.2, 6.3.3, 6.3.4, 6.4.3, 6.4.5, 6.4.7, 6.5.1, 6.5.2, 6.5.3, 6.6.5 (14 problems)

F  4/19 7.2,  7.3, 7.4 (definition and properties of the integral)
HW 12 (due 4/26): 7.2.5, 7.2.6, 7.3.2 (can you make the sequence increasing?), 7.4.6

Exam 3: Monday, 4/22 (topic: Chapter 6). Keywords and results for this part of the course (study guide):

uniform convergence of a sequence of functions (on a given set)- equivalent criterion based on sup |f_n-f|
continuity of the uniform limit
Ascoli-Arzela theorem
Condition for differentiability of the limit
Uniform convergence of a series of functions/ Weierstrass M-test
Convergence interval of a power series/ smoothness and uniform convergence on compact sub-intervals
Taylor series (of a smooth function, at zero): Lagrange's formula for the error

Exam 3
Exam 3 solutions

W 4/24 Fundamental Theorem of Calculus
HW 12: 7.5.4, 7.5.10

F 4/26 Sets of measure zero, Lebesgue's criterion for integrability
Problems discussed: 7.2.5, 7.2.6, 7.3.2 (on Wed.) 7.4.1
Review problems from Chapter 7: 7.3.4,  7.3.5, 7.4.4, 7.4.6, 7.5.4, 7.5.7, 7.5.9, 7.5.10 (Show that F(x) is Lipschitz)

Keywords and results for integration

Definition of the Riemann integral
Integrability of continuous functions, and functions with finite discontinuity set
Example of non-integrable function
Integrability of f  compared to integrability of |f|
Uniform convergence and integrability
The Fundamental Theorem of Calculus (2 statements and two counterexamples)
Sets of measure zero, Lebesgue's criterion for integrability.

Final Exam: Wednesday, May 1st., 10:15-12:15, Ayres 121.

Structure of the final:
six questions taken from Exams 1,2,3 (two from each) and two problems on integration (Chapter 7),
taken from the 12 problems listed above (see lecture of 4/26). (Statements of theorems or definitions are also possible questions.)

Review session: Tuesday, April 30, 5:00-7:00PM, Ayres 111
(the emphasis will be on the eight problems listed as review problems for Chapter 7, see above.)