# Riemann derivative

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.