Symmetric derivative

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 ,

if

and one of order ,

if

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

References

[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

Comments

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