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.