634 
MATHEMATICS: G. A. BLISS 
Then a necessary condition that the sums 5 have a Hnait as 5 approaches 
zero is that 
lim 2) Oi\^iu \ =0, (2) 
5=0 i=\ 
where Oi is the oscillation of /(x) on the interval Xi-iXi. For it is clear 
that the values Xi can always be selected for two sums S, S' on the same 
partition so that the difference S — S' is> as near as is desired to the 
sum in (2). Hence if for every norm 5 a partition can be found with 
n 
^Oi\^iU\>7)>^, 
i= 1 
then for every b two sums S, S' can be found with | 5 — 5' | > 77, and 
this would contradict the existence of the limit (1). 
Let denote the closed set of points x where the oscillation 0{x) of 
/ satisfies the inequality 0{x) ^ e, and let the intervals of S which con- 
tain points of as interior points be designated by the subscript j. 
Then it is also necessary that 
lim S I A,- 1 - 0. (3) 
5 = 0 j 
For in every such interval Oj ^ e, and if for every b a sum as in (3) 
exists exceeding 77, a sum as in (2) can be found exceeding er). 
Consider now a function u(x) which is continuous as well as of lim- 
ited variation, and suppose that the integral (1) exists. Then if 5 
is sufficiently small the ^-intervals of 6* will be such that 
S|A,-^^|<r7. 
/ 
Since u{x) is now continuous the sum S can be altered by the introduc- 
tion of new division points so that all the points of D are interior to 
the set of points defined by the sum of the 7-intervals and the last 
inequality still true. Furthermore for the altered S it will follow that 
2 a} Z7 < 271 
j 
provided that b is chosen so small that the total variation U{h) — U (a) , 
n 
and \ AiU \y and hence also the corresponding quantities for 
every sum of partial intervals of ab, differ by less than 77. Such a 
choice is always possible when u{x) is continuous (Vallee Poussin, 
Coiirs Analyse Infinitesimale, vol. 1, 3d ed., p. 73). It follows then 
