Dini derivative

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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 [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.


[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)


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:
This article was adapted from an original article by T.P. Lukashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article