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 . 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.
|||U. Dini, "Grundlagen für eine Theorie der Funktionen einer veränderlichen reellen Grösse" , Teubner (1892) (Translated from Italian)|
|||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 , , , .
Dini derivative. T.P. Lukashenko (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Dini_derivative&oldid=13670