The four derivates of a function f(x) are defined as follows.
The upper right derivate
The lower right derivate
The upper left derivate
The lower left derivate
Infinite values (positive or negative) are possible. A finite derivative exists at every point x where the four derived numbers have the same finite value.
In a paper in 1908 in the Proceedings of the London Mathematics Society, William Young proved that if f is a continuous function, then f+(x) = f–(x) and f–(x) = f+(x) everywhere except at a set of points of the first category.
Grace Young's 1914 Acta Mathematica paper [Abstract] contains the result that for an arbitrary function f, f+(x) ≥ f–(x) and f–(x) ≥ f+(x) everywhere except at a countable set of points.
Further results appeared in her 1916 Gamble Prize paper in the Quarterly Journal. At the same time, Arnaud Denjoy was studying the same problem in France. Both independently established the following theorem for continuous functions. In a 1916 paper in the London Mathematical Society Proceedings [Abstract], Young proved it for measurable functions. Stanislaw Saks eventually removed the measurability assumption in 1924.
Denjoy-Young-Saks Theorem: Except at a set of measure 0, the derivates of an arbitrary function f(x) at any point x belong to one of the following three cases. Either
- all four derivates are equal and the function is differentiable at x, or
- the upper derivates on each side of x are +∞ and the lower derivates on each side are –∞, i.e.
f–(x) = f+(x) = +∞ and f–(x) = f+(x) = –∞, or
- the upper derivate on one side is +∞, the lower derivate on the other side is –∞ and the two remaining extreme derivates are finite and equal, i.e.
f+(x) = f–(x) are finite, f–(x) = +∞, f+(x) = –∞, or
f–(x) = f+(x) are finite, f+(x) = +∞, f–(x) = –∞.