Solutions to Homework 2
February 2, 2009
18.4:
Let
S
⊆
R
and suppose
∃
a sequence (
x
n
) in
S
that converges to a number
x
0
/
∈
S
. Show
that there
∃
an unbounded continuous function on
S
.
Note: In other words, we want to show that there is a function
f
defined on
S
(and thus defined
on (
x
n
)) such that it is unbounded as
x
approaches
x
0
. With the hint and this property, a good
choice will be
f
(
x
) =
1
x

x
0
Also note, it’s good practice to show that the
f
(
x
) chosen is continuous
everywhere but at
x
0
, but it wasn’t marked wrong if you didn’t prove it.
Proof: Let
f
(
x
) =
1
x

x
0
, and the sequence (
x
n
) be in
S
and converge to
x
0
. Since
f
(
x
) is continuous
on
R
−
x
0
so it is continuous on
S
. lim
n
→∞
f
(
x
n
) = lim
n
→∞
1
x
n

x
0
. Note that for arbitrarily small
ǫ >
0
∃
N >
0 s.t.

x
n
−
x
0

< ǫ
which implies
1
ǫ
<
1

x
n

x
0

. Thus for arbitrarily large
M
= 1
/ǫ
there
exist an
N >
0 (namely the same one above) s.t.

f
(
x
n
)

> M
thus
f
(
x
n
) diverges to infinity as
x
n
→
x
0
, meaning
f
(
x
) is unbounded in
S
.
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 Spring '09
 Continuous function, Xn, intermediate value property

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