View source for Denjoy theorem on derivatives

<!--
d0310401.png
$#A+1 = 12 n = 0
$#C+1 = 12 : ~/encyclopedia/old_files/data/D031/D.0301040 Denjoy theorem on derivatives
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

The Dini derivatives (cf. [[Dini derivative|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 [[#References|[1]]]. The theorem, cf. [[#References|[2]]], which follows is a generalization of Denjoy's theorem: For almost-all  $  x $
the [[Contingent|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Denjoy,   "Mémoire sur les nombres dérivés des fonctions continues"  ''J. Math. Pures Appl. (7)'' , '''1'''  (1915)  pp. 105–240</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Saks,   "Theory of the integral" , Hafner  (1952)  (Translated from French)</TD></TR></table>

====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 [[#References|[a2]]]. She then extended it to measurable  $  F $[[#References|[a3]]]. S. Saks extended the theorem to arbitrary functions  $  F $[[#References|[a1]]].

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