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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:
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<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>
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<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>
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The Dini derivatives (cf. [[Dini derivative|Dini derivative]]) of any finite function  $  F $
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at almost any point  $  x $
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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/d0310405.png" /></td> </tr></table>
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$$
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{\overline{F}\; } {}  ^ {+} ( x)  = {\overline{F}\; } {}  ^ {-} ( x)  = + \infty ,\ \
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{\underline{F} } {}  ^ {+} ( x)  = {\underline{F} } {}  ^ {-} ( x)  = - \infty ;
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$$
  
<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/d0310406.png" /></td> </tr></table>
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$$
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{\overline{F}\; } {}  ^ {+} x  = {\underline{F} } {}  ^ {-} ( x)  \neq  \infty ,\  {F
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under } {}  ^ {+} ( x)  = - \infty ,\  {\overline{F}\; } {}  ^ {-} ( x)  =  + \infty ;
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$$
  
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).
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$$
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{\underline{F} } {}  ^ {+} ( x)  =  {\overline{F}\; } {}  ^ {-} ( x)  \neq  \infty ,\  {F
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bar } {}  ^ {+} ( x)  =  + \infty ,\  {\underline{F} } {}  ^ {-} ( x)  =  - \infty ;
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$$
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 +
$$
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{\overline{F}\; } {}  ^ {+} ( x)  =  {\underline{F} } {}  ^ {+} ( x)  =  {
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\overline{F}\; } {}  ^ {-} ( x)  =  {\underline{F} } {}  ^ {-} ( x)  \neq  \infty .
 +
$$
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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 $  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====
 
====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>
 
<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====
 
====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]]].
+
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====
 
====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>
 
<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>

Latest revision as of 17:32, 5 June 2020


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
How to Cite This Entry:
Denjoy theorem on derivatives. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Denjoy_theorem_on_derivatives&oldid=46625
This article was adapted from an original article by T.P. Lukashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article