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k
(
m
arealso in this intrval.
a
c
m
/2
k
d
(
m
1)/2
a
1/2
.
Check that
b
(
c
)
1/3)/2
L
(
c
),
b
(
d
)
L
(
d
) and
b
(
k
)
L
(
k
)
2/(3.2
), from (vii) and (viii).
(x) Let
be the linear (straight line) function which coincides
with
L
L
on theintrval
m
2
,
m
1
2
[
c
,
d
]
.
By definition
L
(
k
)
L
(
a
)
k
a
L
(
c
)
L
(
a
)
.
c
a
Usethis to show that
b
(
k
)
b
(
a
)
k
a
b
(
c
)
b
(
a
)
c
a
b
(
k
)
L
(
k
)
k
a
,
provided
b
(
a
)
L
(
a
). Show further that
b
(
k
)
L
(
k
)
k
a
(2/3)2
(4/3)2
,
using (ix).
(xi) Usethenotation in (x) to construct a similar proof in thecase
L
(
a
)
b
(
a
).
Start with
L
L
L
(
k
)
k
a
L
(
a
)
(
d
)
(
a
)
.
d
a
Show that
b
(
k
)
b
(
a
)
k
a
b
(
d
)
b
(
a
)
c
a
b
(
k
)
L
(
k
)
,
k
a
and as before show that
b
(
k
)
L
(
k
)
k
a
(2/3)2
(4/3)2
.
(xii) Use (x) and (xi) to show that in every neighbourhood of
a
,
b
(
x
)
b
(
a
)
x
a
takes values which differ by
or more and therefore
b
(
x
)
b
(
a
)
x
a
cannot havea limit as
x
a
.