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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
 
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