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Continuity Notions for Multi-Valued Mappings with Possibly Disconnected Images

We discuss properties of continuity notions for multi-valued mappings which allow disconnected images, but still have a useful zero property. The starting point is the notion of c-continuity introduced by the second author in the book Interval Methods for Systems of Equations to study enclosure properties of interval Newton operators. It was claimed in that book that c-continuity possesses a zero property. However, we provide a counterexample. Two other continuity notions are introduced and examined, and applied in a logical context.