Colloquium publications.

58

THE CAMBRIDGE COLLOQUIUM.

value of n. A function of finite variation may be written as the
difference of two non-decreasing functions
(13') a(x) = p(x) - n(x),
in which, for definiteness, we write p(a) = ac(a). If p(x) and
n(x) are, for each value of x, the least possible functions so definable, the function
(14) t(x) = p(x) - p(a) + n(x)
is called the total variation of a(x), and is the upper limit of the
sum (13) formed for the interval ax, instead of the interval ab.
We may if we like in (13') replace the two non-decreasing
functions by functions which actually increase without remaining
constant in any interval, although for these functions the equation (14) will no longer hold.
On account of (13') it is immediately seen that the discontinuities which a function of finite variation may have are
limited in nature. In fact if x' is a point of discontinuity of
a(x), both of the limits a(x' + 0) and a(x' - 0) must exist if x'
is an interior point of the interval, and one of them, if x' is an
end point of the interval. Moreover, the number of these socalled discontinuities of the first kind is restricted; they must be
denumerable, though not necessarily countable in some preassigned order, say from left to right.
The number of intervals through which a function may remain constant is restricted in the same way.
By means of a function <o(x) which is continuous, and a function a(x) which is of finite variation, in the interval a -- x b,
form the expression
n+l
(15) Zic(hi){a(x i)-a(xi) },
where the points xi are the. same as before, and the points ~i are
arbitrary also except for the restrictions xi_-i - <- x_. The
expression (15) approaches a limit as n becomes infinite, and
the maximum sub-interval approaches zero; this limit is called
the Stieltjes integral and written