Honors Introduction to Analysis I
Homework VII
Solution
March 30, 2009
Problem 1
Let
f
be a monotone function on an interval. Show that if the image of
f
is an interval, then
f
is
continuous.
Give an example of a nonmonotone function on an interval whose image is an interval, but is not
continuous.
Solution.
Since it is defined on an interval
I
, by Theorem 4.2.6,
f
has at most countably many number of points
where it has jump discontinuities and perhaps at the endpoints, it may have lim
x
→
x
+
0
f
(
x
) or lim
x
→
x

0
f
(
x
) equal to
±∞
. Since the image is an interval, the last part is not possible, meaning the limits are finite for the endpoints too.
Wlog assume
f
is increasing and that it has at least one dicontinuity, i.e. there is
x
0
∈
I
such that lim
x
→
x

0
f
(
x
) =
a
and lim
x
→
x
+
0
f
(
x
) =
b
and wlog
a < b
. This means that any value
c
∈
(
a, b
) is not in the image of
f
: if it were,
then we would have
c
=
f
(
y
), for some
y
∈
I
for which we can assume
y > x
0
(the case
y < x
0
is similar). Take
z
∈
(
x
0
, y
) and by monotonicity of
f
, lim
x
→
x
+
0
f
(
x
)
≤
f
(
z
)
≤
f
(
y
) so
b
≤
f
(
z
)
≤
f
(
y
) =
c
, contradicting
c < b
. So
(
a, b
) is not in the image, hence the image cannot be an interval.
For the example, take
f
: [

1
,
1]
→
[0
,
1] with
f
(
x
) =
(
x
x
∈
[0
,
1]
1
x
∈
[

1
,
0)
. The image of [

1
,
1] is [0
,
1], but it is
not continuous at 0.
Problem 2
If
f
is a continuous function on a compact set, show that either
f
has a zero, or
f
is bounded away
from zero (

f
(
x
)

>
1
/n
for all
x
in the domain, for some
1
/n
).
Solution.
The image of a continuous function on a compact set is a compact set, which is closed and bounded.
If there is 0 in the image, than we are done. Assume
f
(
x
)
6
= 0, for any
x
, but it is not bounded away from zero,
i.e.
∀
n
there is
x
n
such that

f
(
x
n
)

<
1
/n
. This means that the sequence
{
f
(
x
n
)
}
has 0 as a limit point. But the
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 '08
 PROTSAK
 Metric space, Cantor set, Cantor, compact set, Archimedean

Click to edit the document details