Qualifying Exam in Analysis August 2009
Instructions: Work ALL the following six problems. Write clearly on one side of your paper
and write your name on every sheet that you use.
Remarks: None of the problems require the use of results from measure theor
SOLUTION OF SELECTED PROBLEMS FROM HOMEWORK SET
VI
Problem 11.1.2 a) /page 3911
a) We have for (x, y ) = (0, 0)
fx (x, y ) =
2x5 + 4x3 y 2 2xy 4
,
(x2 + y 2 )2
and fx (0, 0) = 0. Since
|x5 | < (x2 + y 2 )5/2 , |x3 y 2 | (x2 + y 2 )5/2 , |xy 4 | (x2 + y 2
SOLUTION OF SELECTED PROBLEMS FROM HOMEWORK SET
V
Problem 10.3.9 /page 360
If f : R1 R1 is continuous, then f 1 (I ) is open for every open (this is a general theorem for each open set, in particular for I ).
Assume now that f 1 (I ) is open for each I op
SOLUTION OF SELECTED PROBLEMS FROM HOMEWORK SET
IV
Problem 9.4.8 /page 327
Let x E \ D. By the density of D, there exists cfw_xl D, xl x. By the
uniform continuity of f , cfw_f (xl ) is Cauchy sequence. Indeed, let > 0 and let > 0,
so that f (x) f (y ) <
SOLUTION OF SELECTED PROBLEMS FROM HOMEWORK SET
III
Problem 9.1.8 /page 307
a) If Br (a) \ cfw_a E is innite, then Br (a) \ cfw_a E = , so this direction is
trivial. Conversely, assume that Br (a) \ cfw_a E = for every r > 0. Then pick
xk B 1 (a) \ cfw_a
SOLUTION OF SELECTED PROBLEMS FROM HOMEWORK SET
II
Problem 8.3.7 /page 296
a) Assume that A B is not connected. Then, there exist open sets U, V , so that
A B U V , U V = and (A B ) U = , (A B ) V = . By the connectivity
of A, it follows that either A U o
SOLUTION OF THE EXTRA CREDIT PROBLEMS
Problem 1/Aug. 2005
Clearly, for every x = 0, we have
1
= 0,
n 1 + nx2
while for x = 0, we have f (0) = limn fn (0) = 1. Hence
lim
f (x) =
0 x=0
1 x = 0.
Next, the limit is uniform on [1, ), since
1
.
n
The limit is n
DEPARTMENT OF MATHEMATICS
UNIVERSITY OF KANSAS
Final MATH 766 - Spring 2011
Your Name:
1
(85)
2
(85)
3
(85)
4
(85)
5
(85)
6
(85)
Total (500)
2
(1) (85 points)
Let cfw_U be an open cover of a compact set K in a metric space (X, ).
Show that there is > 0,
DEPARTMENT OF MATHEMATICS
UNIVERSITY OF KANSAS
MIDTERM MATH 766 - Spring 2011
Your Name:
1
(75)
2
(75)
3
(75)
4
(75)
Total (250)
2
(1) (75 points)
Prove that every bounded connected set E of R1 is in one of the forms (a, b),
(a, b], [a, b), [a, b] for som
Luke Hunter
Geo 100
11 November 2016
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