Porosity and the Darboux property of Fréchet derivatives
type: Item
instance of: scholarly article
title: Porosity and the Darboux property of Fréchet derivatives
zbMATH Open document ID: 1185.26019
author: Ondřej Kurka
author: Dušan Pokorný
published in: Real Analysis Exchange
publication date: 2010-01-04T00:00:00Z
review text: Two theorems are proved which are converses in the case of Euclidean spaces of existing theorems in Banach spaces. The new theorems are: Theorem 1.3. Let \(B\subset \mathbb R^d\) be nonempty without isolated points such that the interior of \(B\) is connected. Then the following assertions are equivalent: (i) \(\mathbb R^d\backslash B\) is porous at every \(a\in B\cup\partial B\). (ii) The graph of \(g\) is connected whenever \(g\) is a Fréchet derivative of a function \(f: B\to \mathbb R\) on \(B\). (iii) \(g(B)\) is connected whenever \(g\) is a Fréchet derivative of a function \(f: B\to \mathbb R\) on \(B\). Proposition 4.2. Let \(a\in \mathbb R^d\) and \(E\subset \mathbb R^d\backslash\{a\}\) be a set which is not porous at \(a\). Then there is a Lipschitz function \(f: \mathbb R^d\to \mathbb R\), Fréchet differentiable on \(\mathbb R^d\backslash E\), such that \(f^{\prime}(a)=0\) and \(|f^{\prime}(u|\geq1\) for any \(u\in \mathbb R^d\backslash(E\cup \{a\})\).
reviewed by: Gerald A. Heuer
Mathematics Subject Classification ID: 26B05
zbMATH DE Number: 5652539
zbMATH Keywords: Fréchet derivative
zbMATH Keywords: Darboux property
zbMATH Keywords: porosity
MaRDI profile type: Publication
full work available at URL: realanalexch.34.2.0319
OpenAlex ID: W1964352576
DOI: 10.14321/REALANALEXCH.34.2.0319
description: scientific article; zbMATH DE number 5652539