548 
DERIVATES, EXTREMES, VARIATION 
If x is not a terminal point, it lies in a sequence of intervals 
K>K> ••• 
belonging to the 1°, 2° ••• division of 51. 
Let , 
(a 
n+1 
)• 
Since F(x) is continuous, there exists an s, such that 
\F(x)-F(a m , n )\< e -, m>s (8 
for any x in 8 m . As F m (x) is monotone in S OT , 
| F m (^x~y X TO (A mra ) | | L m (jx mn ^ F m (yd m ^| 
<| F m ( ^mn ) F m (jx m ^ n +i) | 
<| v by 8). 
Thus \L m (x)-F m {a„)\< e ~. (9 
Hence from 8), 9), 
j F(x) — L m (x) | < e , m>s 
which is 7). 
8. We can write 7) as a telescopic series. For 
L 1 = L 9 + (X x — X 0 ) 
X 2 = Xj + (X 2 — Xj) = X 0 + (X x — X 0 ) + (X 2 — X x ) 
etc. Hence 
F(x) = lim Ln(x') = L 0 (x) + f ¿X n (>) - L n _ x (x)\. 
/„(a;) = -E 0 O) , /„(a) = L n (x) - (10 
i(*) = !/„(*), (11 
0 
z;(a0 = £/,(*) = £»(*)• ( 12 
0 
The function /„(#), as 10) shows, is the difference between the 
ordinates of two successive polygons X n _ x , X n at the point #. It 
may be positive or negative. In any case its graph is a polygon 
we have 
and