View source for Denjoy theorem on derivatives

The Dini derivatives (cf. [[Dini derivative|Dini derivative]]) of any finite function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031040/d0310401.png" /> at almost any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031040/d0310402.png" /> satisfy one of the following relations:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031040/d0310403.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031040/d0310404.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031040/d0310405.png" /></td> </tr></table>

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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031040/d0310407.png" /> the [[Contingent|contingent]] of the graph of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031040/d0310408.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031040/d0310409.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031040/d03104010.png" />, independently of Denjoy by G.C. Young [[#References|[a2]]]. She then extended it to measurable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031040/d03104011.png" /> [[#References|[a3]]]. S. Saks extended the theorem to arbitrary functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031040/d03104012.png" /> [[#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>