Peano derivative

One of the generalizations of the concept of a derivative. Let there exist a such that for all with one has

where are constants and as ; let . Then is called the generalized Peano derivative of order of the function at the point . Symbol: ; in particular, , . If exists, then , , also exists. If the finite ordinary two-sided derivative exists, then . The converse is false for : For the function

one has , but does not exist for (since is discontinuous for ). Consequently, the ordinary derivative does not exist for .

Infinite generalized Peano derivatives have also been introduced. Let for all with ,

where are constants and as ( is a number or the symbol ). Then is also called the Peano derivative of order of the function at the point . It was introduced by G. Peano.