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Peano derivative

From Encyclopedia of Mathematics
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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.

How to Cite This Entry:
Peano derivative. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Peano_derivative&oldid=49520
This article was adapted from an original article by A.A. Konyushkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article