Theorems relating to perfectly general functions are rare. In the present connexion there is the theorem, given by myself in the Acta, that the points, if any, at which the upper derivate on one side is less than the lower derivate on the other side, form a countable set. In this theorem no restriction whatever is laid on the primitive function, except tacitly that it is everywhere finite, and the derivates may be ordinary or taken with respect to any monotone increasing function g(x) without constant stretches.
I propose now to enunciate and prove three fundamental theorems, already known to be true for special classes of functions, concerning the derivates of a function f(x), which, if not perfectly general, is nevertheless, for mathematical purposes, practically unrestricted in character. I only assume that
(a) f(x) is a measurable function; and
(b) f(x) is finite everywhere.
The enunciations of these theorems are as follows:—
THEOREM 1.—The points at which the upper derivate on one side is +∞ and the lower derivate on the other side is not –∞, form a set of content zero.
In other words, if f+(x) = +∞, then f–(x) = –∞, except at a set of content zero; and if f–(x) = +∞, then f+(x) = –∞, except at a set of content zero.
THEOREM 2.—The points at which f(x) has an infinite right-hand (left-hand) differential coefficient form a set of content zero.
As an immediate corollary we have the generalisation of Lusin's theorem:—
The points, if any, at which a measurable function f(x) has an infinite differential coefficient, form a set of zero content.
THEOREM 3:—The points at which one of the upper derivates and one of the lower derivates of f(x) are finite and not equal form a set of content zero.
In other words the four sets at which
(i) f+(x) and f–(x), (ii) f+(x) and f+(x), (iii) f–(x) and f–(x), (iv) f–(x) and f+(x),
have finite unequal values, are all sets of zero content.
The three main theorems enable us to state for any measurable function which is finite everywhere the striking properties enunciated for a continuous function by Denjoy:—
If we neglect a set of points whose content is certainly zero, the derivates of f(x) at any point x belong to one of the three following cases:—
Each of these cases may occur at points of a set of any content, as is shown by examples, and all three cases may present themselves simultaneously for a single function f(x), or one or more of the cases may be altogether absent.