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Math 310 
hw
7 solutions
17.4, 17.9d, 17.10c;
Wednesday, 28 Oct 2009
17.4 Prove that the function
x
7→
√
x
is continuous on its domain [0
,
∞
).
Hint:
Apply Example 5 in
§
8.
Let
x
0
∈
[0
,
∞
) and let (
x
n
)
n
be a sequence in [0
,
∞
) such that lim
n
→∞
x
n
=
x
0
. Then by Example 5 in
§
8,
lim
n
→∞
√
x
n
=
√
x
0
.
Hence, the function is continuous at
x
0
by Deﬁnition 17.1. Since
x
0
∈
[0
,
∞
) was chosen arbitrarily, the function
x
7→
√
x
is continuous on its domain [0
,
∞
).
±
17.9d Prove that the following function is continuous at
x
0
by verifying the
ε

δ
property of Theorem 17.2.
g
(
x
) =
x
3
,
x
0
arbitrary.
Hint:
x
3

x
3
0
= (
x

x
0
)(
x
2
+
xx
0
+
x
2
0
).
Let
ε >
0 and let
x
0
be given. Set
M
= 1 +

x
0

. If

x

x
0

<
1 we have:

x
0

,

x

< M
and
x
2
0
, x
2
,

xx
0

< M
2
.
So we set
δ
:= min
(
1
,
ε
3
M
2
)
. Let
x
∈
R
and suppose that

x

x
0

< δ
; then

x

x
0

<
1 and so the above
inequalities hold and therefore:

x
3

x
3
0

=

x
2
+
xx
0
+
x
2
0

x

x
0
 ≤
(
x
2
+

xx
0

+
x
2
0
)

x

x
0

<
3
M
2
δ
≤
ε.
Hence,

x
3

x
3
0

< ε
and so
g
is continuous at
x
0
by Theorem 17.2.
±
17.10c Prove that the function sgn (deﬁned below) is discontinuous at the point
x
0
= 0. You may use either Deﬁnition
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This note was uploaded on 03/26/2010 for the course MATHEMARIC 131a taught by Professor Janeday during the Spring '10 term at San Jose State.
 Spring '10
 janeday

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