Continuity Notions for Multi-Valued Mappings with Possibly Disconnected Images

If we regard a multi-valued mapping as a generalization of a singlevalued mapping, we may ask for generalizations of important properties as well, concerning zeros, fixed points, projection, composition etc. The main interest in this paper are continuity notions for multi-valued mappings which allow disconnected images, but still have a certain zero property. The starting point is a continuity notion introduced by the second author in [23], which was stated to possess a zero property. However, we provide a counterexample to this theorem. Two other continuity notions are introduced and examined, and applied in a logical field.