Peano derivative

One of the generalizations of the concept of a derivative. Let there exist a $\delta > 0$ such that for all $t$ with $| t | < \delta$ one has

$$f( x _ {0} + t) = \alpha _ {0} + \alpha _ {1} t + \dots + \frac{\alpha _ {r} }{r!} t ^ {r} + \gamma ( t) t ^ {r} ,$$

where $\alpha _ {0} \dots \alpha _ {r}$ are constants and $\gamma ( t) \rightarrow 0$ as $t \rightarrow 0$; let $\gamma ( 0) = 0$. Then $\alpha _ {r}$ is called the generalized Peano derivative of order $r$ of the function $f$ at the point $x _ {0}$. Symbol: $f _ {(} r) ( x _ {0} ) = \alpha _ {r}$; in particular, $\alpha _ {0} = f( x _ {0} )$, $\alpha _ {1} = f _ {(} 1) ( x _ {0} )$. If $f _ {(} r) ( x _ {0} )$ exists, then $f _ {(} r- 1) ( x _ {0} )$, $r \geq 1$, also exists. If the finite ordinary two-sided derivative $f ^ { ( r) } ( x _ {0} )$ exists, then $f _ {(} r) ( x _ {0} ) = f ^ { ( r) } ( x _ {0} )$. The converse is false for $r > 1$: For the function

$$f( x) = \left \{ \begin{array}{ll} e ^ {- 1/x ^ {2} } , & x \neq 0 \textrm{ and } \textrm{ rational } , \\ 0, & x = 0 \textrm{ or } \textrm{ irrational } , \\ \end{array} \right .$$

one has $f _ {(} r) ( 0) = 0$, $r = 1, 2 \dots$ but $f _ {(} 1) ( x)$ does not exist for $x \neq 0$( since $f( x)$ is discontinuous for $x \neq 0$). Consequently, the ordinary derivative $f ^ { ( r) } ( 0)$ does not exist for $r > 1$.

Infinite generalized Peano derivatives have also been introduced. Let for all $t$ with $| t | < \delta$,

$$f( x _ {0} + t) = \alpha _ {0} + \alpha _ {1} t + \dots + \frac{\alpha _ {r} ( t) }{r!} t ^ {r} ,$$

where $\alpha _ {0} \dots \alpha _ {r-} 1$ are constants and $\alpha _ {r} ( t) \rightarrow \alpha _ {r}$ as $t \rightarrow 0$( $\alpha _ {r}$ is a number or the symbol $\infty$). Then $\alpha _ {r}$ is also called the Peano derivative of order $r$ of the function $f$ at the point $x _ {0}$. It was introduced by G. Peano.