552 
MATHEMATICS: R. D. CARMICBAEL 
viously a lower bound to St^'. Hence if the number of divisions of 
{ah) is increased by repartitions in such wise that 5 approaches zero, the 
sum 5^/ approaches a definite finite limit. Let us ask under what 
conditions we shall be led to a contradiction by supposing that two 
convergent sequences of sums o-^ for sequences of partitions tt (whether 
formed by repartitions or not) with norms 5 approaching 0 lead to 
different limits Ni and N2 where Ni<N2. Let ^ and rj be two arbitrar- 
ily small positive numbers such that Ni ^ -\- r) < N2. Let xi be a 
partition of (ab) into 5 intervals belonging to the sequence by which 
Ni is defined and let 5 be so great that S^^.^ < Ni -\- ^. Let ^2 be a 
partition of (ab) into t intervals where t is an integer greater than s 
and subject to being made as large as one pleases. Let tts be a par- 
tition formed by the points of tti and 7r2, so that tts is a repartition of 
both TTi and 7r2. Then we have 
s-k 
where the intervals fe^-i, ^'j^), obviously at most 5 — 1 in number, 
are all the intervals of ^2 which are separated into parts in forming X3. 
Since / may be made large at our choice and since the sum in the second 
member of the foregoing relation never has more than s — 1 terms, 
whatever the value of /, it is clear that t may be chosen so large that this 
sum is less than rj provided that u(x) is continuous at the discontinuities 
of f(x). Then we have S^^^ ^ 5^, + 77. But we have seen that 8^3 ^ 
S^, <Ni ^. Hence we have < iVi -f- ^ + rj. But, as / increases, 
approaches N2' Hence we have N2 ^ Ni -i- ^ + rj, contrary to the 
hypothesis iVi + ^ + '7<A^2. Hence the upper integral of f(x) as to a 
mono tonic non-decreasing function u(x) exists provided that u(x) is 
continuous at the points of discontinuity of f(x). The corresponding 
result may likewise be proved for the case of a non-decreasing function 
ti(x); and also for the case of the lower integral. Hence, since every 
function of bounded variation u{x) may be expressed as the difference 
of two monotonic non-decreasing functions which are continuous at the 
points of continuity of u(x), we have the following theorem: 
Theorem I. // u(x) is of bounded variation and f(x) is bounded on 
(ab) and if f(x) is continuous at the points of discontinuity of u(x), then 
the limits as 5 approaches 0 of the sums 5^, o-^ of the preceding paragraph 
both exist. 
Let us now write the function u{x) in the form 
