View source for Symmetric derived number

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''at a point  $  x $''

A generalization of the ordinary notion of a derived number (cf. [[Dini derivative|Dini derivative]]) to the case of a set function  $  \Phi $
on an  $  n $-
dimensional Euclidean space. The symmetric derived numbers of  $  \Phi $
at  $  x $
are defined as the limits

$$ 
\lim\limits _ {k \rightarrow \infty } \ 

\frac{\Phi ( S ( x, r _ {k} )) }{| S ( x, r _ {k} ) | }
 ,
$$

where  $  S ( x, r _ {k} ) $
is some sequence of closed balls with centres at  $  x $
and radii  $  r _ {k} $
such that  $  r _ {k} \rightarrow 0 $
as  $  k \rightarrow \infty $.

The  $  n $-
th symmetric derived numbers at  $  x $
of a function  $  f $
of a real variable are defined as the limits

$$ 
\lim\limits _ {k \rightarrow \infty } \ 

\frac{\Delta _ {s}  ^ {n} f ( x, h _ {k} ) }{h _ {k}  ^ {n} }
 =
$$

$$ 
= \ 
\lim\limits _ {k \rightarrow \infty }  
\frac{\sum _ {m = 0 } ^ { n }  \left ( \begin{array}{c}
n \\
 m 
\end{array}
 \right ) (- 1)  ^ {m} f \left
( x + {
\frac{n - 2m }{2}
 } h _ {k} \right ) }{h _ {k}  ^ {n} }
 ,
$$

where  $  h _ {k} \rightarrow 0 $
as  $  k \rightarrow \infty $
and  $  \Delta _ {s}  ^ {n} f ( x, h _ {k} ) $
is the [[Symmetric difference of order n|symmetric difference of order  $  n $]] 
of  $  f $
at  $  x $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Saks,   "Theory of the integral" , Hafner  (1937)  (Translated from French)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,   "Real and complex analysis" , McGraw-Hill  (1974)  pp. 24</TD></TR></table>