Denjoy theorem on derivatives

The Dini derivatives (cf. Dini derivative) of any finite function $ F $ at almost any point $ x $ satisfy one of the following relations:

$$ {\overline{F}\; } {} ^ {+} ( x) = {\overline{F}\; } {} ^ {-} ( x) = + \infty ,\ \ {\underline{F} } {} ^ {+} ( x) = {\underline{F} } {} ^ {-} ( x) = - \infty ; $$

$$ {\overline{F}\; } {} ^ {+} x = {\underline{F} } {} ^ {-} ( x) \neq \infty ,\ {F under } {} ^ {+} ( x) = - \infty ,\ {\overline{F}\; } {} ^ {-} ( x) = + \infty ; $$

$$ {\underline{F} } {} ^ {+} ( x) = {\overline{F}\; } {} ^ {-} ( x) \neq \infty ,\ {F bar } {} ^ {+} ( x) = + \infty ,\ {\underline{F} } {} ^ {-} ( x) = - \infty ; $$

$$ {\overline{F}\; } {} ^ {+} ( x) = {\underline{F} } {} ^ {+} ( x) = { \overline{F}\; } {} ^ {-} ( x) = {\underline{F} } {} ^ {-} ( x) \neq \infty . $$

The theorem has been demonstrated by A. Denjoy for continuous functions [1]. The theorem, cf. [2], which follows is a generalization of Denjoy's theorem: For almost-all $ x $ the contingent of the graph of $ F $ at a point $ ( x, F( x)) $ is one of the following figures: a plane, a half-plane (with a non-vertical boundary line) or a straight line (non-vertical).

References

[1]	A. Denjoy, "Mémoire sur les nombres dérivés des fonctions continues" J. Math. Pures Appl. (7) , 1 (1915) pp. 105–240
[2]	S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French)

Comments

The theorem cited is often called the Denjoy–Young–Saks theorem. It was discovered and proved, for continuous $ F $, independently of Denjoy by G.C. Young [a2]. She then extended it to measurable $ F $[a3]. S. Saks extended the theorem to arbitrary functions $ F $[a1].

References

[a1]	S. Saks, "Sur les nombres derivées des fonctions" Fund. Math. , 5 (1924) pp. 98–104
[a2]	G.C. Young, Quart. J. Math , 47 (1916) pp. 148–153
[a3]	G.C. Young, "On the derivatives of a function" Proc. London Math. Soc. (2) , 15 (1916) pp. 360–384