Perfect roads for multivalued maps
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.
scientific article; zbMATH DE number 5071497