Geodesic
Dense Continuity and Selections of Set-Valued Mappings
Serdica Mathematical Journal, Tome 24 (1998) no. 1, pp. 49-72.

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A theorem proved by Fort in 1951 says that an upper or lower semi-continuous set-valued mapping from a Baire space A into non-empty compact subsets of a metric space is both lower and upper semi-continuous at the points of a dense Gδ -subset of A. In this paper we show that the conclusion of Fort’s theorem holds under the weaker hypothesis of either upper or lower quasi-continuity. The existence of densely defined continuous selections for lower quasi-continuous mappings is also proved.
Keywords: Set-Valued Mappings, Selections, Semi-Continuity, Quasi-Continuity, Generic, Baire Category 
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Kenderov, Petar; Moors, Warren; Revalski, Julian. Dense Continuity and Selections of Set-Valued Mappings. Serdica Mathematical Journal, Tome 24 (1998) no. 1, pp. 49-72. http://geodesic.mathdoc.fr/item/SMJ2_1998_24_1_a6/