Graphics Reference
In-Depth Information
Answers
1 (i)
R
0
. (ii) Yes. (iii) Yes. (iv) [0, 1) for example. (v) (
,
) for
example. (vi) Yes.
2 If
f
(
x
)
f
(
y
) when
x
y
, thefunction
f
is said to be decreasing or
monotonic decreasing.
If
f
(
x
)
y
, the function is said to be strictly decreasing
or strictly monotonic decreasing.
f
(
y
) when
x
3 (i) R
0
. (ii) Yes. (iii) R
. (iv) Yes.
4 sup
V
exists, by qn 4.80. By 4.64, given
0, for some
x
,
sup
V
f
(
x
)
sup
V
, but
f
is monotonic increasing so, for
x
x
,
sup
V
f
(
x
)
f
(
x
)
sup
V
,so
f
(
x
)
sup
V
and lim
f
(
x
)
sup
V
.
5
f
(
a
) is an upper bound for
L
and a lower bound for
U
, so sup
L
and
inf
U
exist by qns 4.80 and 4.81. Given
0, for some
x
a
,
sup
L
f
(
x
)
sup
L
. But
f
is monotonic increasing so, if
x
x
a
, sup
L
f
(
x
)
f
(
x
)
sup
L
, and
f
(
x
)
sup
L
.
Similarly, for some
x
a
, inf
U
f
(
x
)
inf
U
. But
f
is monotonic
increasing so, if
a
x
x
, inf
U
f
(
x
)
f
(
x
)
inf
U
, and
f
(
x
)
inf
U
.
lim
[
x
]
n
1 and lim
[
x
]
n
.
6 If
f
is monotonic decreasing then, with the notation of qn 5,
f
(
a
)isa
lower bound for
and an upper bound for
U
. Proceed as in qn 5 with
appropriate inequalities reversed.
L
7 If
f
is continuous at
a
, then limits from above and below at
a
areequal.
sup
L
f
(
a
)
inf
U
so, if sup
L
inf
U
, then lim
f
(
x
)
f
(
a
)
lim
f
(
x
),
and so
f
is continuous by qn 6.89.
If sup
inf
U
, then there is a rational number between these two by
qn 4.6. Because the function is monotonic this locates a distinct
rational number in each discontinuity. The open intervals (sup
L
L
, inf
U
),
at thepoints of discontinuity, aredisjoint bcausethefunction is
monotonic. A set of rationals is countable, so the set of discontinuities
of a monotonic function is countable.
8 (i)
1
, (ii) [
1, 1], (iii) (
,
), (iv) (0, 1], (v)
R
, (vi)
R
0
, (vii)
R
.
9 Make your own sketches without considering possible formulae.
