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4.2 Properties of continuity
65
4.2 Properties of continuity
Theorem 4.2.1.
If
f,g
are continuous, then so are
f
+
g
,
f
·
g
, and (if
g
6
= 0
)
f/g
.
Proof.
sequences.
NOTE: Deﬁne (
f
+
g
)(
x
) :=
f
(
x
) +
g
(
x
), etc.
USE this with the thm that lim commutes with continuous functions:
Example 4.2.1.
NOTE: all limits are ﬁnite, and the denom
6
= 0!
lim
t
→
1
3
t
2

√
t
t
2
+ 1
=
lim
t
→
1
(3
t
2

√
t
)
lim
t
→
1
(
t
2
+ 1)
=
3lim
t
→
1
t
2

lim
t
→
1
√
t
lim
t
→
1
t
2
+ lim
t
→
1
1
=
3

√
lim
t
→
1
t
1 + 1
=
2
2
= 1
Theorem 4.2.2.
Let
x
=
g
(
t
)
,c
=
g
(
b
)
. If
g
(
t
)
is continuous at
b
and
f
(
x
)
is continuous
at
c
, then
f
◦
g
(
t
) =
f
(
g
(
t
))
is continuous at
b
.
Proof.
Given
ε >
0
,
∃
δ >
0 such that
f
(
x
)
≈
ε
f
(
c
) for
x
≈
δ
c
continuity of
f
g
(
t
)
≈
δ
g
(
b
) for
t
≈
α
b
continuity of
g.
Then
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This note was uploaded on 11/26/2011 for the course MAT 4944 taught by Professor Wong during the Fall '11 term at FSU.
 Fall '11
 Wong
 Continuity, Limits

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