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Homework 9
H´
ector Guillermo Cu´
ellar R´
ıos
May 9, 2006
23.3 Determine which of the following continuous functions are uniformly con
tinuous on the given set. Justify your answers.
(a)
f
(
x
) =
e
x
x
on [2
,
5]
It is uniformly continuous since it is continuous on a compact set.
(c)
f
(
x
) =
x
2
+ 3
x

5 on [0
,
4] It is uniformly continuous since it is
continuous on a compact set.
(f)
f
(
x
) =
1
x
2
on (0
,
∞
) It is not uniformly continuous since it can
not be extended to a function that is continuous on [0
,
∞
] because
lim
x
→
0
f
(
x
) =
∞
(g)
f
(
x
) =
xsin
(
1
x
) on (0
,
1) It is uniformly continuous since it can be
extended to a function that is contiuous on [0
,
1] by Example 21.5.
23.4 (a)
f
(
x
) =
x
3
on [0
,
2]
Proof.
Given any
ε >
0 we want to make

f
(
x
)

f
(
y
)

< ε
. Now

f
(
x
)

f
(
y
)

=

x
3

y
3

=

x

y

x
2
+
xy
+
y
2

but

x
2
+
xy
+
y
2
 ≤
12, then we let
δ
=
ε
12
and

x

y

< δ
, we have

f
(
x
)

f
(
y
)

=

x

y
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This note was uploaded on 01/27/2010 for the course MATHM 413 taught by Professor Michaeljolly during the Spring '08 term at Indiana.
 Spring '08
 MichaelJolly

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