Symmetric differentiability of functions of two variables
type: Item
instance of: scholarly article
zbMATH Open document ID: 1006.26005
author: M. Okropiridze
publication date: 2003-03-09T00:00:00Z
Mathematics Subject Classification ID: 26B05
zbMATH DE Number: 1586386
zbMATH Keywords: symmetric differentiability
zbMATH Keywords: symmetric strong partial derivatives
MaRDI profile type: Publication
title: Symmetric differentiability of functions of two variables
published in: Proceedings of A. Razmadze Mathematical Institute
review text: A function \(f: \mathbb{R}\times \mathbb{R}\to \mathbb{R}\) is said to be symmetrically differentiable at the point \((x_0, y_0)\) if there exist finite numbers \(A= A(x_0, y_0)\) and \(B= (x_0, y_0)\) such that the equality NEWLINE\[NEWLINE\lim_{(h,k)\to (0,0)} (f(x_0+ h,y_0+ k)- f(x_0- h,y_0- k)- 2Ah- 2Bk)/(|h|+|k|)= 0NEWLINE\]NEWLINE is fulfilled. The author investigates this notion in connection with notions of symmetric partial derivatives in usual, in strong and in the angular sense. E.g., it is proved: If there exist symmetric strong partial derivatives \(f^{(')}_{(1)}(x_0, y_0)\) and \(f^{(')}_{(2)}(x_0, y_0)\), then \(f(x,y)\) is symmetrically differentiable at \((x_0, y_0)\) and not vice versa (a function \(f(x,y)\) is said to possess at the point \((x_0, y_0)\) a symmetric strong partial derivative with respect to \(x\) if there exists the limit NEWLINE\[NEWLINE\lim_{(h,k)\to (0,0)} (f(x_0+ h,y_0+ k)- f(x_0- h,y_0+ k))/2h= f^{(')}_{(1)}(x_0, y_0);NEWLINE\]NEWLINE the symmetric strong partial derivative with respect to \(y\), \(f^{(')}_{(2)}(x_0, y_0)\), is defined analogously).
reviewed by: Pavel Kostyrko
description: scientific article