# 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 , , , .

**How to Cite This Entry:**

Dini derivative. T.P. Lukashenko (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Dini_derivative&oldid=13670