Dini derivative

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derived numbers

A concept in the theory of functions of a real variable. The upper right-hand Dini derivative is defined to be the limes superior of the quotient as , where . The lower right-hand , the upper left-hand , and the lower left-hand Dini derivative are defined analogously. If (), then has at the point 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 , , , .

How to Cite This Entry:
Dini derivative. T.P. Lukashenko (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098