Dini derivative
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.
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 ,
,
,
.
Dini derivative. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dini_derivative&oldid=13670