636 
MATHEMATICS: G. A. BLISS 
is expressible as a sum u{x) = v{x) -\-j{x), where v{x) is of limited 
variation and continuous, and7(^) is the 'function of jumps' 
iW= 2 [«©-«'(«-o)]+ S t«(« + o)-«(i)l- (5) 
The sums on the right are taken in each case for all the discontinuities ^ 
satisfying the inequality below the summation sign, and are absolutely 
convergent. 
It can be shown first of all that if the integral (1) exists the discon- 
tinuities of / and u are necessarily distinct. For suppose there were 
a discontinuity ^ common to both with \u{^ + 0) — — 0)1 > rj, 
0{^) > v- On a sufficiently small interval ^— h^x^^ + h 
the absolute value of -\- h) — — h) and the oscillation of / 
would both exceed 77. Hence for every norm 5 there would exist two 
sums S and S' coinciding except on this interval and such that 
15 — 5"! > 1?^, which contradicts the existence of the limit (1). If 
— 0) = + 0) at a discontinuity ^, sums contradicting the 
existence of the limit can be similarly constructed by using intervals 
having ^ as end point. 
In the second place the integral (1) with 7 (^) in place of u{x) always 
exists if the discontinuities of / and u are distinct, and it has the 
value 
the sum being taken for all discontinuities ^ of u{x) with the under- 
standing that u{a — 0) = u{a), u(b -f 0) = u{b). The series (5) for 
X = b converge absolutely, and it is possible to select a set of discon- 
tinuities (k =1,2, . . . , w) so that the sum of the absolute values 
of the terms of (5) not involving the ^kS is less than e, and the corre- 
sponding sum from (6) less than eM. Consider now a sum S for the 
integral (6) with norm 5 so small that no two values ^k can lie in the 
same interval. The value of 5 is the sum of the terms oij{b) from (5) 
each multiplied by a factor of the form f{X) with |X — ^ | < 5. The 
terms not involving the '^kS have a sum less than eM in absolute value, 
while those involving the ^kS will differ from the corresponding terms 
of (6) by less than eM, say, if d is sufficiently small since / is continuous 
at every ^. Hence for a sufficiently small 8, the sum 5 differs from the 
sum on the right in (6) by less than 3 eM, which proves the statement 
at the beginning of this paragraph. 
It follows then that a necessary and sufficient condition for the in- 
tegral (1) to exist is that / and u have no discontinuity in common, and 
a<^<x 
