Graphics Reference
In-Depth Information
10 Take
A
as the union of (two) disjoint and separated intervals.
11 Thefunction is continuous by qn 6.29. Thenumbr 0 is not in the
range.
12 If
f
(
d
)
0, then either
f
(
d
)
0or
f
(
d
)
0. If
f
(
d
)
0, then take
a
a
and
b
d
.If
f
(
d
)
0, take
a
d
and
b
b
. Now suppose
f
(
a
)
0
0 and let
d
b
and
f
(
b
)
(
a
). If
f
(
d
)
0 wehavefinished. If
a
d
.If
f
(
d
)
d
and
f
(
d
)
0, take
a
and
b
0, take
a
b
b
.
13
(i) (
a
) is monotonic increasing and bounded above by
b
, and thus is
convergent, by qn 4.35. (
b
) is monotonic decreasing and bounded
below by
a
, and thus is convergent, by qn 4.34.
(ii) (
f
(
a
))
f
(
A
) by thecontinuity of
f
and
f
(
A
)
0 by qn 3.78, the
closed interval property.
(iii) From qns 12 and 3.54(v),
B
A
(
B
A
), so
B
A
.
f
(
A
)
0
f
(
A
)
f
(
A
)
0.
14 Define
g
(
x
)
f
(
x
)
k
, then
g
(
a
)
0 and
g
(
b
)
0 and
g
is continuous
by qn 6.23. So
g
(
c
)
0 for some
c
(
a
,
b
), and
f
(
c
)
k
0, so
f
(
c
)
k
.
15 Define
g
(
x
)
f
(
x
), then
g
(
a
)
k
g
(
b
) and
g
is continuous, by qn
6.26. So
g
(
c
)
k
for some
c
(
a
,
b
), and
f
(
c
)
k
.
16 The following function reaches every value between 0 and 1, but is
discontinuous everywhere.
f
: [0, 1]
[0, 1].
f
(0)
;
f
(
)
0;
f
(
x
)
x
when
x
is rational and
0,
;
f
(
x
)
x
when
x
is irrational.
Also, the function of qn 6.19 reaches every value between
f
(0) and
f
(
x
),
but is discontinuous at 0.
1
17 If
f
(
a
)
f
(
b
) but both are integers then, by qn 14, for some
c
(
a
,
b
),
f
(
c
)
f
(
a
)
which is not an integer.
18 If
f
(
a
)
f
(
b
) but both arerational then, by qn 14, for some
c
(
a
,
b
),
f
(
a
)
2
f
(
b
)
1
2
f
(
c
)
which is irrational. See qn 4.20.
19
(i) If
x
a
b
1, then
f
(
x
)
0. If
x
a
b
1, then
f
(
x
)
0
by the Intermediate Value Theorem. This result may be extended
to any polynomial of odd degree.
0. So there is a real root of the equation
x
ax
b
(ii)
f
(0)
0,
f
(
a
)
0, when
a
1, so by the Intermediate Value
Theorem there is a root
in [0,
a
].
a
0, so
a
.If
a
1, apply I.V.T. to [0,1].
