Perfect roads for multivalued maps
type: Item
instance of: scholarly article
title: Perfect roads for multivalued maps
zbMATH Open document ID: 1110.54012
author: Joanna Czarnowska
published in: Real Analysis Exchange
publication date: 2006-11-13T00:00:00Z
review text: The following theorem of Maximoff is extended to a multivalued case: ``Let \(f:\mathbb R \to \mathbb R\) be a first Baire class function (\(f^{-1}(V)\) is an \(F_\sigma\)-set for any open \(V\subset \mathbb R\)). Then \(f\) has the Darboux property if and only if \(f\) has a perfect road at each point.'' The author defines upper (lower) first class multivalued maps as well as upper (lower) perfect roads, and proves, as a consequence of several intermediate results, that each compact valued both upper and lower first class multivalued map \(F: \mathbb R\multimap \mathbb R\) with a lower and an upper perfect road at each point has the intermediate value property. Note that this property implies the Darboux property if values of \(F\) are connected.
reviewed by: Grzegorz Gabor
Mathematics Subject Classification ID: 54C60
zbMATH DE Number: 5071497
zbMATH Keywords: Darboux property
zbMATH Keywords: perfect road
MaRDI profile type: Publication
full work available at URL: realanalexch.31.2.0365
OpenAlex ID: W1598194135
DOI: 10.14321/REALANALEXCH.31.2.0365
description: scientific article; zbMATH DE number 5071497