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''Schwarzian derivative, second symmetric derivative, of a function  $  f $
at a point  $  x _ {0} $''

The limit

$$ 
D  ^ {2} f( x _ {0} )  = \ 
\lim\limits _ {h \rightarrow 0 }  
\frac{f( x _ {0} + h) - 2f( x _ {0} ) + f( x _ {0} - h) }{h  ^ {2}
 }
 .
$$

It was introduced by B. Riemann in 1854, who proved that if at a point  $  x _ {0} $
the second derivative  $  f ^ { \prime\prime } ( x _ {0} ) $
exists, then so does the Riemann derivative and  $  D  ^ {2} f( x _ {0} ) = f ^ { \prime\prime } ( x _ {0} ) $. 
The upper and lower limits of

$$ 

\frac{f( x _ {0} + h) - 2f( x _ {0} ) + f( x _ {0} + h) }{h  ^ {2} }

$$

as  $  h \rightarrow 0 $
are called the upper ( $  {\overline{D}\; } {}  ^ {2} f( x _ {0} ) $) 
and lower ( $  \underline{D}  ^ {2} f( x _ {0} ) $) 
Riemann derivative, respectively.

Riemann derivatives find wide application in the theory of the representation of functions by trigonometric series, and in particular in connection with the [[Riemann summation method|Riemann summation method]].

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.M. Apostol,   "Mathematical analysis" , Blaisdell  (1957)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B. Riemann,   "Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe" , ''Gesammelte Math. Abhandlungen'' , Dover, reprint  (1957)  pp. 227–264</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Wolff,   "Fourier'sche Reihen" , Noordhoff  (1931)</TD></TR></table>