Graphics Reference
In-Depth Information
R
R
x
27 Why does the function
f
:
0
defined by
f
(
x
)
havean
inverse whether the positive integer
n
is odd or even?
Say why there is a continuous inverse
f
:
R
0
R
0
. This
inverse function is normally denoted by
x
x
or
x
.
We established the existence of
n
th roots for positive real numbers in
qn 4.40. The argument here establishes that the function
x
x
is
continuous on R
0
.
28 Find thelimit
1
x
1
x
lim
,
when
m
and
n
are positive integers with
m
n
. (Another method is
given in qn 9.26.)
Continuous functions on a closed interval
29 What does the Intermediate Value Theorem allow us to claim
about thepossiblerangs of a continuous function
f
:[
a
,
b
]
R?
30 Attempt to draw graphs of continuous functions
f
:[
a
,
b
]
with
the seven different ranges as in qn 9. Use pencil and paper, not a
computer.
R
We now investigate whether the fact that the ranges found in qn 30
were all bounded above and bounded below, is a necessary consequence
of theconditions, or an indication of our lack of imagination. We
suppose that there exists a continuous function
f
:[
a
,
b
]
R whose
range is unbounded above and see whether any contradiction arises.
31 Wesupposethat therangeof thecontinuous function
R
is unbounded above. The unboundedness of the range of
f
means
that, whatever integer
n
wechoos, wecan find an
x
in [
a
,
b
] such
that
f
(
x
)
n
. If such an
x
can befound wecall it
x
f
:[
a
,
b
]
. This gives us
a sequence (
x
).
(i) Is there any reason why the sequence (
x
) should be
convergent?
(ii) Is the sequence (
x
) bounded?
(iii) Must the sequence (
x
) contain a convergent subsequence?
(iv) If the subsequence (
x
) converges to
c
, why must
c
[
a
,
b
]?
(v) What does the continuity of
f
allow us to say about the
sequence (
f
(
x
))?
(vi) For suMciently large
n
can webesurethat
f
(
x
)
f
(
c
)
1?
