Graphics Reference
In-Depth Information
Inverses of continuous functions
22 If
f
:[
a
,
b
]
is a continuous and one
—
onefunction with
f
(
a
)
f
(
b
), provethat
f
is strictly monotonic increasing.
Deduce that the range of
f
is [
f
(
a
),
f
(
b
)].
What is theanalogous rsult if
f
(
a
)
f
(
b
)?
R
23 If
f
:[
a
,
b
]
R is strictly monotonic, must
f
be
(i) one
—
one;
(ii) invertible;
(iii) continuous?
R
24 If
f
:[
a
,
b
]
is continuous and invertible (see qn 6.6) must
f
be
strictly monotonic?
25 Supposethat
f
:[
a
,
b
]
R is a continuous and strictly monotonic
function,
(i) How do you know that therangeof
f
is a closed interval
[
c
,
d
], say?
(ii) How do you know that
f
has an inverse function
g
:[
c
,
d
]
[
a
,
b
]?
(iii) How do you know that
g
must bestrictly monotonic?
(iv) Let
x
beany point in thedomain of
f
and let
y
f
(
x
), so that
a
x
b
and
c
y
d
.
How do you know that lim
g
(
t
) and lim
g
(
t
) exist?
(v) Let (
x
)
x
be an increasing sequence in [
a
,
b
] and let (
s
)
x
be a decreasing sequence in [
a
,
b
]. Why must (
f
(
x
))
f
(
x
)
f
(
x
), and of these two sequences why must one
be increasing and one decreasing?
(vi) Deduce that there is a sequence (
t
and (
f
(
s
))
y
from below and one
tending to
y
from abovefor each of which (
g
(
t
)
))
g
(
y
).
(vii) Why does this establish that
g
is continuous at
y
?
Wehavenow proved that a function which is continuous and strictly
monotonicon an interval has a continuous inverse.The use of completeness
in qn 25 is unavoidable. Consider
f
: Q
Q given by
f
(
x
)
x
when
x
2
and
f
(
x
)
x
. An example of a one
—
onecontinuous
function Q
Q, whose inverse is discontinuous everywhere, is given in the
topic by Dieudonne´ cited in the bibliography.
x
1 when
2
26 Does the function
f
: R
R defined by
f
(
x
)
x
have an inverse? Find a
maximal interval
A
R, with 1
A
for which thefunction
f
:
A
R
defined by
f
(
x
)
x
is monotonic. With this
A
, is thefunction
f
:
A
A
a bijection? Is the function
f
:
A
A
continuous? How is it usually
denoted?
