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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 .
 
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