Ukrainian Mathematical Journal, Vol.69, No.9, February, 2018 (Ukrainian Original Vol.69, No.9, September, 2017) POINTS OF UPPER AND LOWER SEMICONTINUITY FOR MULTIVALUED FUNCTIONS A. K. Mirmostafaee UDC 515.12 We investigate joint upper and lower semicontinuity of two-variable set-valued functions. More precisely, among other results, we show that, under certain conditions, a two-variable lower horizontally quasicon- tinuous mapping F : X ⇥ Y ! K(Z) is jointly upper semicontinuous on sets of the form D ⇥ {y0}, where D is a dense G δ -subset of X and y0 2 Y. A similar result was obtained for the joint lower semi- continuity of upper horizontally quasicontinuous mappings. These results improve some known results on the joint continuity of single-valued functions. 1. Introduction and Preliminaries Throughout the paper, we assume that all topological spaces are T 1 . For a topological space Z, by P (Z ), C (Z ), and K(Z ) we denote the set of all nonempty subsets, the set of all nonempty closed subsets, and the set of all nonempty compact subsets of Z, respectively. Let F : X ! P (Z ) be a set-valued function. For a subset G of Z, we define F + (G) and F − (G) as follows: F + (G)= {x 2 X : F (x) ✓ G}, F − (G)= {x 2 X : F (x) \ G 6= ∅}. The function F is called: (a) upper (resp., lower) semicontinuous if, for every open subset G of Z, F + (G) (resp., F − (G) ) is an open subset of X. (b) upper (resp., lower) quasicontinuous at x 0 2 X if for any open set G with x 0 2 F + (G) (resp., x 0 2 F − (G) ) and any neighborhood U of x 0 , there exists a nonempty open set V ✓ U such that V ✓ F + (G) (resp., V ✓ F − (G) ). (c) categorically upper (resp., lower) quasicontinuous at x 0 2 X if, for each neighborhood U of x 0 and a neighborhood G containing F (x 0 ), there exists a set A ✓ U of the second category such that F (a) ✓ G (resp., F (a) \ G 6= ∅) for all a 2 A. Let Z be a topological space and let {G n } be a sequence of open covers of Z. For every z 2 Z and n 2 N, let St(z, G n )= [ {G 2 G n : z 2 G n }. The sequence {G n } is called: (a) A development if, for every z 2 Z, the sequence {St(z, G n )} is a base at z. A space with a developable space is called a developable space. A regular developable space is called a Moore space. Ferdowsi University of Mashhad, Mashhad, Iran. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 69, No. 9, pp. 1224–1231, September, 2017. Original article submitted Novem- ber 12, 2014; revision submitted February 8, 2017. 1424 0041-5995/18/6909–1424 c � 2018 Springer Science+Business Media, LLC DOI 10.1007/s11253-018-1441-z