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Math 341 Homework # 8
P104. 13, 14, 17. P105. 19, 20, 21, 22.
13. Let
f
:
D
→
R
be continuous at
x
0
∈
D
. Prove that there is
M >
0 and
a neighborhood
Q
of
x
0
such that

f
(
x
)
 ≤
M
for all
x
∈
Q
∩
D
.
Proof: Since
f
is continuous at
x
0
,
∀
± >
0,
∃
δ >
0, for
x
∈
D
with

x

x
0

< δ
, we have

f
(
x
)

f
(
x
0
)

< ±.
Let
±
= 1,
M
=

f
(
x
0
)

+ 1 and
Q
= (
x
0

δ,x
0
+
δ
). Then for
x
∈
Q
∩
D
,

f
(
x
)

=

f
(
x
)

f
(
x
0
) +
f
(
x
0
)

≤ 
f
(
x
)

f
(
x
0
)

+

f
(
x
0
)

<
1 +

f
(
x
0
)

=
M.
14. If
f
:
D
→
R
is continuous at
x
0
∈
D
, prove that

f

:
D
→
R
such that

f

(
x
) =

f
(
x
)

is continuous at
x
0
.
Proof: Since
f
is continuous at
x
0
,
∀
± >
0,
∃
δ >
0, for
x
∈
D
with

x

x
0

< δ
, we have

f
(
x
)

f
(
x
0
)

< ±.
Hence

f

(
x
)
 
f

(
x
0
)
 ≤ 
f
(
x
)

f
(
x
0
)

< ±.
Thus,

f

is continuous at
x
0
.
17. Suppose
f
:
D
→
R
with
f
(
x
)
≥
0 for all
x
∈
D
. Show that, if
f
is
continuous at
x
0
, then
√
f
is continuous at
x
0
.
Proof: Since
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This note was uploaded on 08/13/2009 for the course MAT 371 taught by Professor Thieme during the Spring '07 term at ASU.
 Spring '07
 thieme
 Math

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