Vol. 35, No. 1, 1970

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Noncontinuous multifuctions

R. Hrycay

Vol. 35 (1970), No. 1, 141–154

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.

Mathematical Subject Classification
Primary: 54.65
Received: 10 October 1969
Revised: 26 February 1970
Published: 1 October 1970
R. Hrycay