# Symmetric derivative

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

$$\lim\limits _ {r \downarrow 0 } \ \frac{\Phi ( S ( x; r)) }{| S ( x; r) | } \equiv \ D _ { \mathop{\rm sym} } \Phi ( x),$$

where $S ( x; r)$ is the closed ball with centre $x$ and radius $r$, if this limit exists. The symmetric derivative of order $n$ at a point $x$ of a function $f$ of a real variable is defined as the limit

$$\lim\limits _ {h \rightarrow 0 } \ \frac{\Delta _ {s} ^ {n} f ( x, h) }{h ^ {n} } =$$

$$= \ \lim\limits _ {h \rightarrow 0 } \frac{\sum _ {k = 0 } ^ { n } \left ( \begin{array}{c} n \\ k \end{array} \right ) (- 1) ^ {k} f \left ( x + { \frac{n - 2k }{2} } h \right ) }{h ^ {n} } = D _ { \mathop{\rm sym} } ^ {n} f ( x).$$

A function $f$ of a real variable has a symmetric derivative of order $2r$ at a point $x$,

$$D _ { \mathop{\rm sym} } ^ {2r} f ( x) = \beta _ {2r} ,$$

if

$${ \frac{1}{2} } ( f ( x + h) + f ( x - h)) - \sum _ {k = 0 } ^ { r } \beta _ {2k} \frac{h ^ {2k} }{( 2k)! } = \ o ( h ^ {2r} );$$

and one of order $2r + 1$,

$$D _ { \mathop{\rm sym} } ^ {2r + 1 } f ( x) = \ \beta _ {2r + 1 } ,$$

if

$${ \frac{1}{2} } ( f ( x + h) - f ( x - h)) - \sum _ {k = 0 } ^ { r } \beta _ {2k + 1 } \frac{h ^ {2k + 1 } }{( 2k + 1)! } = \ o ( h ^ {2r + 1 } ).$$

If $f$ has an $n$- th order derivative $f ^ { ( n) }$ at a point $x$, then there is (in both cases) a symmetric derivative at $x$, and it is equal to $f ^ { ( n) } ( x)$.

#### 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