Dini derivative

From Encyclopedia of Mathematics
Revision as of 17:04, 7 February 2011 by (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

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. 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