Math 508
Exam 2
Jerry L. Kazdan
December 9, 2010
9:00 – 10:20
Directions
This exam has three parts, Part A asks for 3 examples (5 points each, so 15 points).
Part B has 4 shorter problems (8 points each so 32 points). Part C has 4 traditional problems (15
points each so 60 points). Total is 107 points.
Closed book, no calculators or computers– but you may use one 3
×
5
card with notes on both
sides.
Part A: Examples
(3 examples, 5 points each so 15 points). Give an example having the specified
property.
1. A function
f
∈
C
1
([

1
,
1]) but is not in
C
2
([

1
,
1]).
2. A bounded sequence
a
k
in a complete metric space
M
where
a
k
has no convergent subsequence.
3. A sequence of continuous functions
f
n
(
x
)
∈
C
([0
,
1]) that converges pointwise to zero but
1
0
f
n
(
x
)
dx
≥
1. [A clear sketch is adequate.]
Part B: Short Problems
(4 problems, 8 points each so 32 points)
B–1. Let
f
(
x
) be a smooth function with the properties:
f
(

1) = 1,
f
(0) = 0, and
f
(1) = 1.
Show that
f
(
c
) = 2 at some
c
∈
(

1
,
1). [Suggestion: Consider
g
(
x
) :=
f
(
x
)

x
2
.]
B–2. Let
2
x
0
f
(
t
)
dt
=
e
cos(3
x
+1)
+
A
. Find
f
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 Math, Continuous function, Metric space, Compact space, Uniform space, complete metric space

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