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18.100B/C: Fall 2008
Homework 6
Available
Wednesday, October 15
Due
Wednesday, October 22
1. (10pts) Problem 1, page 98 in Rudin
2. (10pts) Problem 3, page 98 in Rudin
3. (10 pts) Let (X, d) be a metric space. Fix x0 X and a continuous function g : R R.
S
18.100B/C: Fall 2008
Homework 7
Available Monday, October 20
Due
Wednesday, October 29
1. Let f : X Y be a continuous function between metric spaces. Dene a function
f : X (0, ) (0, ) as follows:
f (x, ) = supcfw_ > 0 ; t X dX (x, t) < dY (f (x), f (t) <
18.100B/C: Fall 2008
Homework 8
Available Tuesday, October 28
Due
Wednesday, November 5
Turn in the homework by 11am on Wednesday, November 5th, in 2-108. For 18.100B
it should be put in the bin corresponding to the lecture you regularly attend (regardles
18.100B/C: Fall 2008
Homework 9
Available
Monday, November 3
Not Due
If you would like feedback on your solutions, you can turn in the homework by 11am on
Wednesday, November 12, in 2-108.
1. Show that sin(x)
obtain
x is a good approximation for small x b
18.100B/C: Fall 2008
Homework 10
Available Wednesday, November 12
Due Wednesday, November 19
1. Recall that a subset N R is said to have measure 0 if, for each > 0, there is a sequence
(nite or countable) of balls (Bn ) with radii rn so that N n Bn and n
18.100B/C: Fall 2008
Homework 11
Available
Wednesday, November 19
Due
Wednesday, November 26
1. (20 pts) Problem 7, page 138 in Rudin.
2. (10 pts) Problem 8, page 138 in Rudin.
3. Use the two denitions given in the previous two problems to answer the foll
18.100B/C: Fall 2008
Homework 12
Available
Wednesday, November 26
Due Friday, December 5
1.
(a) (10pts) Problem 2, page 165 in Rudin.
(b) (10pts) Problem 3, page 165 in Rudin.
2. (10pts) Show that fn (x) =
1
n
+ x2 converges uniformly on R to f (x) = |x|.
18.100B/C Midterm Exam
Thursday, November 13 2008, 7:309:00, in 1-190.
Closed book, no calculators.
YOUR NAME:
YOUR SECTION (circle one):
18.100B MWF 12-1
18.100B TR 1-2:30
18.100C
This is a 90-minute evening exam. No notes, books, or calculators are perm
1. (15pts) (1) Write down
5
1+2i
in the form of a + bi for a, b R.
sol. 1 2i
(2) Find all solutions z C to the equation |z |2 = 3z 2.
sol. Write z = x + iy . Then, the equation becomes x2 + y 2 =
3x 3iy 2 y = 0 and x2 3x + 2 = 0 z = 1, 2.
2. (25pts) Deter
Solutions of Homework 1
1.
Suppose S is an ordered set with the greatest-lower-bound property, B S , B is not empty,
and B is bounded above. Let L be the set of all upper bound of B . Then, L is not empty
because we assume B is bounded above, hence there
18.100B/C: Fall 2008
Homework 5
Available Wednesday, October 8
Due Wednesday, October 15
Turn in the homework by 11am on Wednesday, October 15, in 2-108. For 18.100B it should
be put in the bin corresponding to the lecture you regularly attend (regardless
18.100B/C: Fall 2008
Homework 4
Available
Tuesday, September 30
Not due
If you would like feedback on your solutions, you can turn in the homework by 11am on
Wednesday, October 8, in 2-108. For 18.100B it should be put in the bin corresponding to
the lect
18.100B/C: Fall 2008
Homework 3
Available
Wednesday, September 24
Due
Wednesday, October 1
Turn in the homework by 11am on Wednesday, October 1, in 2-108. For 18.100B it should
be put in the bin corresponding to the lecture you regularly attend (regardles
p
IS COMPLETE
Let 1 p , and recall the denition of the metric space
p
:
For 1 p < ,
p
=
sequences a = (an ) in R such that
n=1
|an |p < ;
n=1
( an )
n=1
such that supnN |an | < . We
whereas
consists of all those sequences a =
p
dened the p-norm as the fu
18.100B/C FALL 2008: PRACTICE EXAM #1
Problems
1) Let (X, d) be a metric space, x0 X and r > 0. Let
B = B (x0 , r) = cfw_x X/d(x0 , x) < r
and
C = cfw_x X/d(x0 , x) r.
a) Prove that B C .
b) Give an example of a space (X, d), a point x0 and a radius r
18.100B/C FALL 2008: PRACTICE EXAM #2
(1) See problem set 9 for Taylors theorem and Taylor series.
(2) (a) Let f : X R be uniformly continuous. Show that if (an ) is Cauchy
in X , then (f (an ) is convergent in R.
(b) Conversely, suppose f : X R maps Cauc
18.100C Presentation topics for 9/10
Problem 1 (Rudin Problems 4,5 pg.22) Review the notion of upper and lower
bound, inmum, and supremum, then solve:
(i) Let E S be a nonempty subset of an ordered set S . Suppose S is a
lower bound of E and S is an upppe
CONTINUOUS ALMOST EVERYWHERE
Denition 1. Let be a subset of R. We say that has measure 0 if, for each > 0, there
is a sequence of balls (Bj = Brj (cj )j N with radii rj > 0 (and centres cj R), such that
j N Bj and rj < .
j =1
The balls here are open inte
18.100 Practice Midterm Solutions
10/5/2008
Problem 1
a. Let x B . If x B , then d(x0 , x) < r and so x C . Else, if x is a limit point of B , then there
exists a sequence xn x with xn B . Then for every > 0, we can nd N such that for all n N ,
d(xn , x)
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THE WEIERSTRASS PATHOLOGICAL FUNCTION
Until Weierstrass published his shocking paper in 1872, most of the mathematical world
(including luminaries like Gauss) believed that a continuous function could only fail to
be differentiable at some collection of i
18.100B/C: Fall 2008
Homework 1
Available
Monday, September 8
Due
Wednesday, September 17
Problems 14 cover material discussed in the rst week of classes, while problems 59
cover material from the second week. Turn in the homework by 11am on Wednesday,
Se
18.100B/C: Fall 2008
Homework 2
Available Monday, September 15
Due
Wednesday, September 24
Turn in the homework by 11am on Wednesday, September 24, in 2-108. For 18.100B it
should be put in the bin corresponding to the lecture you regularly attend (regard
Solutions of Homework 2
11 in Chapter 1.
By the part (c) of the theorem 1.33, r should be |z | because |z | = |rw| = |r|w| = r. Hence, if
z = 0, r = |z | > 0 and w can be set by |z | . It can be check that z = rw for r = |z | and w = |z |
z
z
using the pa
Solutions of Homework 3
17 in Chapter 2.
Countable: There is a natural one-to-one map from E to [0, 1] by mapping each digit 4 and
7 of x in E to 0 and 1 respectively. Hence, E is not countable because [0, 1] is not countable.
Dense: E is not dense in [0,
Solutions of Homework 4
1 in Chapter 3.
Suppose sn s, hence for any > 0, there exists N such that |sn s| < for n > N .
Because |sn | |s| |sn s| < for n > N , cfw_sn converges to |s|. The converse is not true
when sn = (1)n .
5 in Chapter 3.
sup (an + bn
18.100B : Fall 2010 : Section R2
Homework 7
Due Tuesday, October 26, 1pm
Reading: Tue Oct.19 : continuity, Rudin 4.1-12
Thu Oct.21 : Quiz 2 (covering Rudin sections 2.45-47 and 3), p spaces.
1 . (a) Problem 1, page 98 in Rudin
(b) Problem 3, page 98 in R
18.100B Fall 2010
Practice Quiz 1 Solutions
1.(a) E X is compact if, given any open cover E
U by open sets U X (with
A
A any index set), one can nd a nite subcover E
U , with A A a nite subset.
A
(b) E = cfw_e1 , . . . , eN
Given an open cover E
U ,