Colloquium publications.

102

THE CAMBRIDGE COLLOQUIUM.

For the resolution of (2), we must first define the class L of
functions sp(x).
Represent by ts(xs) the total variation function for a(xs), the
variable x being considered as a parameter. If for a set E of
values of x, or an interval xxz2 of values of x, the quantity t,(xs),
which is of course c ts(xb), remains finite, c T., the function
a(xs) is said to be of uniformly limited variation in s over the set
E, or the interval x1x2, of values of x. In this case the approximation sum approaches its limit uniformly, and the measure of
the approach of the sum to the integral is given by the quantity
co^T, as follows from (16), Lecture III.
We may obtain also directly the inequality
(3) <p(s)dsa(xs) f MT,
where Ip(x) M.
Upon these inequalities is based the following theorem, about
the continuity of Stieltjes integrals:
THEOREM. If in the integral
rb
(4) i4(x) = f (p(s)da(xs), (p(s) continuous, a - s c- b,
ca(xs) is of uniformly limited variation in s for x in the neighborhood
of xo, and if there is a set of values Fxo of s which includes a and b
and is dense in ab, such that ao(xs) is continuous in x at xo, for s in
Fxo, then {/(x) is continuous at XG.*
It is evidently then desirable, for (2), to take as the class L
the class of all functions p(s), continuous a '- s b, in order to
satisfy (i); the condition (ii) will also be satisfied if we take
jX < 1/Ta, as we see by (3). Accordingly, we have the theorem:
THEOREM. If a(xs) is of uniformly limited variation in s for
a c x c b, and if for every value x = Xo in this interval there is a
set Fxo of values of s, including a and b and dense in ab, such that
if s is in Fx, the function a(xs) is continuous in x at xo, then equation (2) has one and only one solution <p(x), continuous a c- x b,
provided that X is small enough ( X I < I/ T).
* This theorem, and the theorems of Art. 62 about the Stieltjes integral,
are proved by H. C. Bray, Annals of Mathematics, vol. 19 (1918).