Graphics Reference
In-Depth Information
(i) How do you know the sequences (
a
) and (
b
) areboth
convergent?
(ii) If (
a
)
A
and (
b
)
B
, why must
f
(
A
)
0
f
(
B
)?
(iii) Why must
A
B
and
f
(
A
)
0?
Wesummarisethersult of qn 13 by saying that, if
f
:[
a
,
b
]
R is a continuous function,
f
(
a
)
0,
f
(
b
)
0, there
exists a
c
(
a
,
b
) such that
f
(
c
)
0.
14 By applying the result of qn 13 to a suitably chosen function, prove
that, if
f
:[
a
,
b
]
R is a continuous function and
f
(
a
)
k
f
(
b
),
then there exists a
c
(
a
,
b
) such that
f
(
c
)
k
.
15 By applying the result of qn 14 to a suitably chosen function, prove
that if
f
:[
a
,
b
]
R
k
f
(
b
),
is a continuous function and
f
(
a
)
then there exists a
c
(
a
,
b
) such that
f
(
c
)
k
.
Questions 14 and 15, together, give the general form of the
Intermediate Value Theorem, that a continuous function
f
:[
a
,
b
]
R
takes every value between
f
(
a
) and
f
(
b
).
R
16 Givean exampleof a discontinuous function
f
:[
a
,
b
]
which
takes every value between
f
(
a
) and
f
(
b
). The existence of such a
function shows that the converse of the Intermediate Value
Theorem (qn 13) is false.
17 Provethat a continuous function
f
: R
Z is necessarily constant.
18 Provethat a continuous function
f
: R
Q is necessarily constant.
19
(i) For any real numbers
a
and
b
show that the function defined
on
x
ax
b
has a real root. To what class of
polynomial functions may this result be extended?
(ii) For any positive real number
a
and integer
n
R
by
f
(
x
)
1, show that
thefunction
f
defined by
f
(
x
)
x
a
has a positiveroot. Use
this to provide an alternative proof of the existence of
n
th
roots for positive numbers to that given in qn 4.40.
20 If
f
: [0, 1]
[0, 1] is a continuous function, by applying thersult
of qn 13 to an appropriately chosen function, prove that there is at
least one
c
[0, 1] such that
f
(
c
)
c
. Givean exampleto show that
this result might fail if the domain of
f
were (0, 1). Generalise this
result for a continuous function
f
:[
a
,
b
]
[
a
,
b
].
21 Can the Intermediate Value Theorem be used to prove that the
image of an interval under a continuous function is necessarily an
interval? Or, in other words, under a continuous function, is a
connected set always mapped onto a connected set?