MATHEMATICS: G. A. BUSS 
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that the total variation of u on the set must be zero, and hence 
also the total variation of u on the totality D of discontinuities of f{x), 
since D is the sum of the sets Z)i/„ {n =1,2, . . . ). 
To prove the sufhciency of the condition of the theorem for a func- 
tion u{x) which is continuous, consider the two monotonically increas- 
ing continuous functions P{x), N(x) satisfying the relations (Op. cit., 
p. 72) 
u{x) - u(a) = P{x) - N(x), U(x) = P{x) -\- N{x). (4) 
If the variation of U{x) is zero on the set D then the same is clearly 
true of P{x) and N(x). Hence it is only necessary to prove the suf- 
ficiency for a monotonically increasing function P(x). If D can be 
enclosed in the interior of a denumerably infinite system of intervals 
on which the sum of the variations of P{x) is less than ri the same is 
clearly true of its sub-set D^, and the latter is furthermore entirely 
interior to a finite number of the intervals since it is closed. (This 
follows from the Heine-Borel theorem. See, for example, Op. cit., p. 
60, Lemme I. The proof is quite similar for a closed set or an interval.) 
The portion of ab remaining can be subdivided into intervals so small 
after is extracted that on each of them the oscillation of f{x) is less 
than 2e since on each such portion 0{x) < e (Op. cit., p. 254, §242). 
For every rj and e a partition of ab is defined in this way for which 
the inequality 
"^OiAiP ^2Mv + 2e[P(b)-P(a)] 
i= 1 
is true, where M is the upper bound of | / 1 onab. The first term on the 
right dominates the part of the sum for which the intervals contain 
points of Z)g, while the second does the same for the remaining terms. 
Hence the lower bound of the sum on the left is zero. But this is a 
sufficient condition that the integral 
exists, as may be proved by precisely the method usually applied to the 
integral of Riemann (Op. cit., pp. 250, 255). It follows with the help 
of (4) that the integral (1) exists since the same is true when u is re- 
placed by either P or iV, and u (x) — u (a) = P — N. 
The only case remaining to be considered is that of a function u(x) 
of limited variation which has discontinuities. It is known that the 
totality of discontinuities of such a function is denumerable, and u(x) 
