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The classes of noncontinuous
multifunctions studied here are characterized by their members having certain
connectedness properties. A particular example is the class of connected
(C,0) multifunctions whose members take connected, open sets to connected
sets. Relationships between these classes are given, and some results known
for connected, single valued functions are generalized to connected (C,0)
multifunctions.
Section 2 contains a continuity theorem for connected (C,0) multifunctions as
well as a necessary and sufficient condition for a topological space to be locally
connected in terms of a condition on the class of connected (C,0) multifunctions
defined on it. In §3, with the aid of the notion of the cluster set of a multifunction at
a point, sufficient conditions are given for a multifunction to be connected. Some
general properties about cluster sets are also proved. Section 4 contains
characterizations of continuity for linear operators, semi-norms and convex
functions.
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