Symmetric derivative

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A generalization of the concept of derivative to the case of set functions on an -dimensional Euclidean space. The symmetric derivative at a point is the limit

where is the closed ball with centre and radius , if this limit exists. The symmetric derivative of order at a point of a function of a real variable is defined as the limit

A function of a real variable has a symmetric derivative of order at a point ,


and one of order ,


If has an -th order derivative at a point , then there is (in both cases) a symmetric derivative at , and it is equal to .


[1] S. Saks, "Theory of the integral" , Hafner (1937) (Translated from French)
[2] R.D. James, "Generalized th primitives" Trans. Amer. Math. Soc. , 76 : 1 (1954) pp. 149–176


In [1] instead of derivative, "derivate" is used: symmetric derivate.

How to Cite This Entry:
Symmetric 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