Colloquium publications.

2

THE CAMBRIDGE COLLOQUIUM.

The argument of a functional, if a curve or a surface, may be
regarded as having sense; and the value of a functional of a
curve will depend therefore generally on the direction in which
the curve is taken, and the value of a functional of a surface
upon which choice we make for the positive aspect of the surface.
A variable y which depends upon all the values of a continuous
function 9p(x), in a range a < x ~ b, that is, by means of a
relation of the form (1), may for many purposes be considered
as a function of an infinite number of variables, e. g., of the values
of sp which correspond to the rational values of x. In fact, since
this infinity is a denumerable one, the properties of functionals
of a continuous function may in a measure be foretold by considering the properties of functions depending upon a finite
number of variables and letting that number become infinite.
As an instance, consider the definition of continuity. A function f(xi... xn) is said to be continuous at a point (x1~ *. Xn~)
if the quantity if - fo I can be made as small as desired by
taking the quantities IXi - xi, i = 1, 2, * * *, n, all less than 5,
with 8 small enough. A similar conception applies to a function
of an infinite number of variables,* and therefore to a functional.
b
We shall therefore say that y[p(x)] is continuous in vo for so = so
~ a
if the limit as 8 vanishes of y[sp(x)] is y[kpo(x)], where | p(x)
- po(x) I < 5, a <x -< b; in other words, as so approaches (o0
uniformly.
2. Derivative of a Functional. The same generalization is not
serviceable directly in the extension of the idea of derivative.
It is therefore from this point of view more convenient to generalize the idea of differential-a linear continuous function of the
increments of the infinite number of variables thus leading to a
linear continuous functional of the increment 0(x) of sp(x). The
functional derivative is however itself a natural conception.
In the neighborhood of a value xo, let <p(x) be given a continuous increment 0(x) which does not change sign: let us write
* It may be desirable to replace, in some cases, the quantity a by iai,
i = 1, 2, * *, n, where the o- are given " scale " values (in accordance with
E. H. Moore's concept of relative uniformity).