/mathcal{F}-Connectivity and Strong /mathcal{F}-Connectivity of Multivalued Maps

Abstract

In the paper the general connectivity property is given for multivalued maps and the Darboux property, the intermediate value property, functional connectivity property, connectivity property etc. are considered as subcases of this property. This general property is characterized locally, so as corollaries we obtain local characterization of the Darboux property, the intermediate value property etc. for multivalued maps and for real functions those classical results given by Bruckner, Ceder [2] and Garret, Nelms and Kellum [5]. Characterization of the sets of Darboux points, the intermediate value property points etc. for multivalued maps and for real functions are straightforward corollaries from one general theorem (Theorem 11).