CM221A
ANALYSIS I
Solutions to Sheet 3
1. Since the sums and products of continuous functions are continuous functions,
it is suﬃcient to prove the statement for
x
n
, where
n
is a nonnegative integer.
Let us ﬁx
c
∈
R
and show that
x
n
is continuous at
c
.
We have
c
n

x
n
= (
x

y
)
P
(
c,x
) where
P
(
c,x
) =
c
n

1
+
c
n

2
x
+
...
+
cx
n

2
+
x
n

1
.
If

c

x

< δ
and
δ <
1 then

x

<

c

+ 1,

P
(
c,x
)

<

c

n

1
+

c

n

2

x

+
...
+

c

x

n

2
+

x

n

1
≤
n
(

c

+ 1)
n

1
and, consequently,

c
n

x
n

< δ n
(

c

+ 1)
n

1
. It follows that

c
n

x
n

< ε
whenever

c

x

< δ
and the positive number
δ
satisﬁes the inequalities
δ <
1
and
δ < εn

1
(

c

+ 1)
1

n
. Thus, for any
ε >
0 we can ﬁnd
δ >
0 such that

x

x

< δ
⇒ 
c
n

x
n

< ε
, which means that
x
n
is continuous at
c
.
2. If
c
∈
(0
,
+
∞
) then

√
x

√
c

=

x

c

(
√
x
+
√
c
)

1
. If
c >
0 then the right
hand side is estimated by
c

1
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 Spring '09
 Topology

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