Math 5319
Exam I
February 12, 2001
Answer the problems on separate paper. You do not need to rewrite the problem statements on
your answer sheets. Do your own work. Show all relevant steps which lead to your solutions.
Retain this question sheet for your records.
In-Class
1.
For each of the following sets (S, G, X, Q) determine whether the sets are:
(a) open;
(d) bounded;
(b) closed;
(e) totally bounded;
(c) connected;
(f) compact.
Also, find the diameter of each set.
i.
S
' { <x , y> 0 R2 : 0 < xy < 1 }
ii.
G
'
iii.
X
' Y c { 2 } c Z , where Y ' {
˜G,
4
n'1
n
Gn
(S d R2)
' (0,1% 1 )
(G d R1)
n
(X d R )
j
n
k '0
1
2k
4
}n ' 0 , and Z
' { 2 n % 1 }4n ' 1
n
1
iv.
Q
'
—Q
4
n'1
2n
, where, for k even, Qk
(Q d R )
' { 1 , 3 , 5 , ã , k&1 }
k k k
k
1
2.
Let <M, > be a metric space. Let A and B be subsets of M. Prove that if A and B are are
compact, then A c B is compact.
3.
Let J be an interval in R1. Let f and g be real-valued, uniformly continuous functions on
J. Prove that f + g is uniformly continuous on J.
4.
Let <M, > be a metric space. Let f be a continuous, real-valued function on M. Prove
that if M is connected, then the image, f (M), is an interval.
5.
Let f be a bounded real-valued function on a closed bounded interval [a, b]. Prove or
disprove each of the following propositions:
(a)
(b)
If f 0 5[a, b], then f 2 0 5[a, b].
If f 2 0 5[a, b], then f 0 5[a, b].
Take-Home (Due Friday, 5:00 pm)
Do five (5) of the following problems:
1.
Let E d R1. Show that if E is uncountable, then there exists a point x0 0 E such that x0 is
a limit point of E.
2.
Let <M, > be a metric space and let x0 0 M. Define f : M 6 R1 by f (x) = (x , x0).
Show that f is uniformly continuous on M.
3.
Let f be a continuous, one-to-one map from R1 into R1. Show that f is a
homeomorphism.
4.
Let f be a bounded real-valued function on a closed bounded interval [a, b].
Let f 0 5[a, b]. Let M ' max f (x) . Show only using material from 7.1-7.3, that
m
b
f
a
5.
#
x 0 [a,b]
m
b
M.
a
Let f (x) = 2 + x on [0,2]. Let
equal width. Find U [ f ;
6.
—Q
n ].
n
Find
be the uniform partition of [0,2] into subintervals of
m
2
f by finding lim U[ f ;
n 64
0
3
n].
' { 1 , 3 , 5 , ã , k&1 }.
Let Q (
'
Let p(x)
' 3456 x 3 & 4272 x 2 % 1412 x . Define f on [0,1] by
f (x)
'
n'1
2n
, where, for k even, Qk
35
35
2n
x 0 Q2n d Q (
p(x)
x Ø Q(
k k k
35
. Find [f ; x] for each x 0 [0,1].
k