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A concept in the theory of functions of a real variable. The upper right-hand Dini derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032530/d0325301.png" /> is defined to be the limes superior of the quotient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032530/d0325302.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032530/d0325303.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032530/d0325304.png" />. The lower right-hand <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032530/d0325305.png" />, the upper left-hand <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032530/d0325306.png" />, and the lower left-hand Dini derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032530/d0325307.png" /> are defined analogously. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032530/d0325308.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032530/d0325309.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032530/d03253010.png" /> has at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032530/d03253011.png" /> a one-sided right-hand (left-hand) Dini derivative. The ordinary derivative exists if all four Dini derivatives coincide. Dini derivatives were introduced by U. Dini [[#References|[1]]]. As N.N. Luzin showed, if all four Dini derivatives are finite on a set, then the function has an ordinary derivative almost-everywhere on that set.
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A concept in the theory of functions of a real variable. The upper right-hand Dini derivative $\Lambda_\alpha$ is defined to be the [[limes superior]] of the quotient $(f(x_1)-f(x))/(x_1-x)$ as $x_1\to x$, where $x_1>x$. The lower right-hand $\lambda_\alpha$, the upper left-hand $\Lambda_g$, and the lower left-hand Dini derivative $\lambda_g$ are defined analogously. If $\Lambda_\alpha=\lambda_\alpha$ ($\Lambda_g=\lambda_g$), then $f$ has at the point $x$ a one-sided right-hand (left-hand) Dini derivative. The ordinary derivative exists if all four Dini derivatives coincide. Dini derivatives were introduced by U. Dini [[#References|[1]]]. As N.N. Luzin showed, if all four Dini derivatives are finite on a set, then the function has an ordinary derivative almost-everywhere on that set.
  
 
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====Comments====
 
====Comments====
The Dini derivatives are also called the Dini derivates, and are frequently denoted also by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032530/d03253012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032530/d03253013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032530/d03253014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032530/d03253015.png" />.
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The Dini derivatives are also called the Dini derivates, and are frequently denoted also by $D^+f(x)$, $D_+f(x)$, $D^-f(x)$, $D_-f(x)$.

Latest revision as of 15:14, 8 May 2017

derived numbers

A concept in the theory of functions of a real variable. The upper right-hand Dini derivative $\Lambda_\alpha$ is defined to be the limes superior of the quotient $(f(x_1)-f(x))/(x_1-x)$ as $x_1\to x$, where $x_1>x$. The lower right-hand $\lambda_\alpha$, the upper left-hand $\Lambda_g$, and the lower left-hand Dini derivative $\lambda_g$ are defined analogously. If $\Lambda_\alpha=\lambda_\alpha$ ($\Lambda_g=\lambda_g$), then $f$ has at the point $x$ a one-sided right-hand (left-hand) Dini derivative. The ordinary derivative exists if all four Dini derivatives coincide. Dini derivatives were introduced by U. Dini [1]. As N.N. Luzin showed, if all four Dini derivatives are finite on a set, then the function has an ordinary derivative almost-everywhere on that set.

References

[1] U. Dini, "Grundlagen für eine Theorie der Funktionen einer veränderlichen reellen Grösse" , Teubner (1892) (Translated from Italian)
[2] S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French)


Comments

The Dini derivatives are also called the Dini derivates, and are frequently denoted also by $D^+f(x)$, $D_+f(x)$, $D^-f(x)$, $D_-f(x)$.

How to Cite This Entry:
Dini derivative. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Dini_derivative&oldid=13670
This article was adapted from an original article by T.P. Lukashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article