On approximate Peano derivatives
type: Item
instance of: scholarly article
title: On approximate Peano derivatives
zbMATH Open document ID: 0813.26001
DOI: 10.1007/BF01876034
author: Hajrudin Fejzić
published in: Acta Mathematica Hungarica
publication date: 1995-05-29T00:00:00Z
review text: A function \(f: \mathbb{R}\to \mathbb{R}\) is said to be \(k\) times approximately Peano differentiable at \(x\in \mathbb{R}\) if there are numbers \(f_ 1(x),\dots, f_ k(x)\) and a set \(V_ x\) of density 1 at \(x\) such that \[ f(x+ t)= f(x)+ {t\over 1!} f_ 1(x)+\cdots+ {t^ k\over k!} f_ k(x)+ t^ k \varepsilon_ k(x,t), \] where \[ \lim_{x+ t\in V_ x,t\to 0} \varepsilon_ k(x, t)= 0. \] The main result of this paper reads as follows: If \(f: \mathbb{R}\to \mathbb{R}\) is \(k\) times approximately Peano differentiable at each \(x\in \mathbb{R}\), then there is a sequence of closed sets \((E_ n)_{n\in \mathbb{N}}\) so that \(\bigcup^ \infty_{n=1} E_ n= \mathbb{R}\) and \(f_{k-1}|_{E_ n}\) is differentiable and \(f_{k-1}^ \prime |_{E_ n} (x)= f_ k(x)\) for each \(n\in \mathbb{N}\). The author gives some applications of this theorem.
Mathematics Subject Classification ID: 26A24
zbMATH DE Number: 638218
zbMATH Keywords: approximate Peano derivatives
zbMATH Keywords: composite derivatives
zbMATH Keywords: path derivatives
zbMATH Keywords: selective derivatives
reviewed by: Andrzej Smajdor
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 638218