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L
16
D
(0)
(
a
) by qn 6.89, the neighbourhood definition of continuity.
17 Expression
A
(
y
)
D
(
x
y
)
A
(
c
) ·
D
(
c
)
M
, sinceboth
A
and
D
are
strictly increasing on
Q
0
.
18
A
is uniformly continuous on
[0,
c
] by qn 7.42, theLipschitz
condition, and therefore may be extended to a continuous function in a
unique way by qn 7.47. This extension may be made at any point
x
Q
0 by selecting
c
x
.
19
A
(
0 for all
x
, thecontinuity of
A
at
x
is
equivalent to thecontinuity of
A
at
x
)
1/
A
(
x
). Since
A
(
x
)
x
from qn. 6.52.
20 If
x
and
y
are rational numbers, these results were obtained in qn 8.
Let (
x
) be a sequence of rational numbers tending to
x
(possibly
irrational) and let (
y
) be a sequence of rational numbers tending to
y
(also possibly irrational), then (
x
y
)
x
y
and (
x
y
)
xy
by qn
3.54 (iii) and (vi). By thecontinuity of
A
,(
A
(
x
))
A
(
x
), (
A
(
y
))
A
(
y
),
(
A
(
x
y
))
A
(
x
y
), (
A
(
xy
))
A
(
xy
) and, by qn 8,
A
(
x
), so by qn 3.54(vi), theproduct rul,
A
(
x
y
)
A
(
x
)·
A
(
y
). So wehavethefirst law. For thescond law,
first establish that it holds for
y
n
N by induction. Then establish
that it holds for
y
y
)
A
(
x
) ·
A
(
y
1/
n
using the bijection of qn 4. Consider a product
of
n
terms each equal to
a
. Then establish that it holds for
y
m
/
n
,a
positive rational, using the two previous cases and extend this to
negative rational
y
from thedefinitions. Wenow havetheequipment
to claim that
A
(
xy
)
(
A
(
x
))
. Now (
A
(
xy
))
A
(
xy
) as wesaw bfore
and ((
A
(
x
))
)
(
A
(
x
))
, by thecontinuity of
A
, substituting
a
for
a
.So
A
(
xy
)
(
A
(
x
))
.
21 Weknow that
A
is strictly increasing on Q and that
A
is continuous
on R.
If (
p
) is an increasing sequence of rationals tending to an irrational
r
,
then by the continuity of
A
,(
A
(
p
))
A
(
r
), and since
A
is increasing,
(
A
(
p
)) is increasing, so by qn 4.66,
A
(
r
)
sup
A
(
p
)
n
N
, and since
every rational
p
r
can belong to a possible (
p
) sequence,
A
(
p
)
p
Q
,
p
r
A
(
q
)
q
Q
,
r
q
A
(
r
)
sup
. Likewise
A
(
r
)
inf
.
If
p
,
q
Q
A
(
q
). Further, for any two
distinct real numbers
r
and
s
, with
r
s
, there exists a rational number
q
such that
r
q
s
, and so
A
(
r
)
and
p
r
q
, then
A
(
p
)
A
(
r
)
A
(
q
)
A
(
s
), and
A
is strictly
increasing on
R
.
22 Since
A
is a bijection R
R
, there is a unique
x
such that
A
(
x
)
X
and a unique
y
such that
A
(
y
)
Y
. From qn 20(i),
A
(
x
y
)
XY
,so
A
(
XY
)
x
y
, and log
XY
log
X
log
Y
.
23
(iii) 0
q
x
q
, and
D
strictly increasing, implies
D
(
q
)
D
(
x
)
D
(
q
). From qn 14, (
D
(
q
))
L
(
a
) and