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Mathematics 250
Fall 2008
Proof Practice # 7
Given open domains
U
⊂
R
m
,
V
⊂
R
‘
, and functions
F
:
U
→
V
and
G
:
V
→
R
k
, show that if
F
and
G
are continuous, then the composition
G
◦
F
is also continuous.
Response: Some of you approached this through the
±

δ
deﬁnition of continuity, and others approached
it through the characterization of continuity by means of inverse images of open sets. Either approach is
viable. We will discuss both.
For the
±

δ
deﬁnition, given a point
p
o
in
U
, and
± >
0, we would want to ﬁnd
δ >
0 such that when

x

p
o

< δ
, we also have

(
G
◦
F
)(
x
)

(
G
◦
F
)(
p
o
)

< ±
. To do this, we ﬁrst use the fact that
G
is
continuous. This means that, for the given
±
and the point
F
(
p
o
), we can ﬁnd
δ
1
>
0 such that if
y
is in
V
and

y

F
(
p
o
)

< δ
1
, then

G
(
y
)

G
(
F
(
p
o
))

< ±
. Then, since
F
is continuous, since
δ
1
>
0, we can ﬁnd
δ >
0 such that if

x

p
o

< δ
, then

F
(
x
)

F
(
p
o
)

< δ
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This note was uploaded on 09/24/2009 for the course MATH 250 taught by Professor Rogerhowe during the Fall '06 term at Yale.
 Fall '06
 RogerHowe
 Math

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