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Tufts University
Department of Mathematics
Math 136 Homework 5
This assignment is due Tuesday, March 11, 2008 in class.
1. (15 points) Let
± >
0. Let
f
:
R
n
→
R
be bounded and let
R
be a nonempty subset of
R
n
. Assume
that for all
x
1
∈
R
and all
x
2
∈
R
,

f
(
x
1
)

f
(
x
2
)

< ±
. Prove that sup
R
f

inf
R
f
≤
±
. This will prove
Theorem 3 on the hints to Homework 4.
2. (25 points) Let
A
⊂
R
n
be bounded and let
f
:
A
→
R
and
g
:
A
→
R
both be bounded and integrable.
As usual, let
f
and
g
denote the extensions to
R
. Let
B
be a rectangle such that
A
⊂
B
.
(a) Let
R
be a nonempty subset of
B
. Prove inf
R
f
+ inf
R
g
≤
inf
R
(
f
+
g
)
≤
sup
R
(
f
+
g
)
≤
sup
R
f
+ sup
R
g
.
(b) Let
± >
0. Prove there is a partition
P
of
B
such that
both
of the following hold for this same
partition
Z
f

±
2
< L
(
f,P
)
≤
U
(
f,P
)
<
Z
f
+
±
2
Z
g

±
2
< L
(
g,P
)
≤
U
(
g,P
)
<
Z
g
+
±
2
(c) Use the result of parts (a) and (b) to prove that
f
+
g
is integrable and
R
A
(
f
+
g
) =
R
A
f
+
R
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This note was uploaded on 03/27/2008 for the course MATH 136 taught by Professor Quinto during the Spring '08 term at Tufts.
 Spring '08
 Quinto
 Math

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