MAT 472 HOMEWORK 9 Due: Tuesday, 11/12/02
Homework is due in my mailboxby 3:00. Write neatly, not too small, and not too lightly.
You may discuss the problems with other students from class, but you must write your
own solutions.
Reread
your proofs
before
copying them out to turn in; I really do mean
that you should write (at least) one draft of each solution. If you submit some nonsense
in the course of a proof, the whole thing might strike me as not worth reading.
38.
Let
g
:[
a; b
]
[
c; d
]
!
R
be continuous, and dene
G
:[
a; b
]
!
R
by
G
(
x
)=
R
d
c
g
(
x; y
)
dy
.Prove that
G
is continuous.
39.
(i) Let
f
k
n
g
1
n
=1
be a sequence of functions on
R
with the following properties:
(a)
k
n
is continuous.
(b)
k
n
0.
(c)
R
1
1
k
n
= 1 (where this means lim
r
!1
R
r
r
k
n
= 1).
(d) For all
r>
0, lim
n
!1
R
r
r
k
n
=1.
Let
f
:[
1
;
1]
!
R
be integrable, and suppose that
f
is continuous at 0. Prove that
lim
n
!1
Z
1
1
f
k
n
=
f
(0)
(ii) Given that
R
1
1
e
x
2
dx
=
p
,prove that
k
n
(
x
)=
r
n
e
nx
2
has properties (a) { (d).
40.
In all parts of this problem, prove that your answer is correct.
(i) Find a sequence of continuous functions
f
n
:
R
!
R
such that
lim
n
!1
lim
x
!
0
f
n
(
x
)
6
= lim
x
!
0
lim
n
!1
f
n
(
x
)
;
and such that both iterated limits exist.
(ii) Find a sequence of continuous functions
f
n
:[0
;
1]
!
R
suchthat
f
n
converges to 0
pointwise, and the sequence of integrals
R
1
0
f
n
does not converge.
41.
In all parts of this problem, prove that your answer is correct.
(i) Find a sequence of Riemann integrable functions on [0
;
1] that converge to a bounded
function that is not Riemann integrable.
(ii) Find a uniformly convergent sequence of dierentiable functions
f
n
:
R
!
R
such
that the sequence
f
0
n
does not converge (p ointwise).
