Math 602 Exam 1 (02/08/11). Solutions
1
x d(x) = 5
1. Give an example of a strictly increasing function : [0, 1] R such that
1
and
0
x2 d(x) = 2.
0
Construction. Recall that
f d is linear with respect to ; it behaves well under addition
or rescaling of .
MAT 602 Homework 4 solutions
# 1. Let X be the interval [0, 1] with the standard metric. Let CR (X ) denote the
subspace of C (X ) that consists of all real-valued continuous functions on X .
(a) Prove that the set A = cfw_f CR (X ) : x X f (x) > 0 is ope
MAT 602 HOMEWORK 2 SOLUTION
# 1. Suppose that : [a, b] R is a strictly increasing function; that is, x > y implies
(x) > (y ). Also, suppose that f : [a, b] [0, ) is a nonnegative continuous function
b
such that a f d = 0. Prove that f (x) = 0 for all a <
MAT 602 HOMEWORK 1
#3 (a), (c), (d) on page 138.
(a) Suppose f (0) = f (0+). Then for any
> 0 there is > 0 such that |f (x)
f (0)| < whenever 0 < x . Let P be the partition cfw_1, 0, , 1. Then
U (P, f, 1 ) = sup f f (0) +
[0, ]
and
L(P, f, 1 ) = inf f f
Math 602 Final Exam (05/11/11). Max total score 50.
Your name:
Out of the seven problems, do any ve (worth 10 points each). You may use your textbook
and notes, but you should work individually.
Due at 4:30pm Wednesday, May 11.
1. Dene f : [0, 1] R as fol
Math 602 Exam 3 (04/05/11). Max total score 50.
Your name:
READ THIS FIRST: Do not open the exam booklet until told to do so. Out of the rst ve
problems, do any four (worth 10 points each). If you attempt all ve problems, indicate which
one is not to be g
Math 602 Exam 3 (04/05/11). Solutions.
1. Suppose that f : [0, 1] R and g : [0, 1] R are continuous functions such that f (x) < g (x) for all
x [0, 1]. Prove that there exists a polynomial P such that f (x) < P (x) < g (x) for all x [0, 1].
Proof. Since g
Math 602 Exam 2 (03/03/11). Max total score 50.
Your name:
READ THIS FIRST: Do not open the exam booklet until told to do so. Out of the rst ve
problems, do any four (worth 10 points each). If you attempt all ve problems, indicate which
one is not to be g
Math 602 Exam 2 solutions.
1. Let F be an equicontinuous family of functions from R to R.
Prove that the family F1 = cfw_f g : f, g F is also equicontinuous.
Proof. Fix
> 0. By the equicontinuity of F , there exists > 0 such that for all f F we have
|f (
Math 602 Exam 1 (02/08/11). Max total score 50.
Your name:
READ THIS FIRST: Do not open the exam booklet until told to do so. Out of the rst ve
problems, do any four (worth 10 points each). If you attempt all four problems, indicate which
one is not to be
MAT 602 HOMEWORK 8 SOLUTIONS
# 2. Suppose that E Rn is open and f : E R is a function such that
the partial derivatives D1 f, . . . Dn f exist and are bounded in E . Prove that f is
continuous in E .
Proof. Let M be an upper bound for partial derivatives.