It is provided a notion of partial lower continuity of setvalued mapping and theorems on automatical partial continuity and selection are proved. It follows from these theorems a generalization of Baire theorem on projection for separately continuous mappings which defined on product of regular strongly countablly complete space and metrizable compact.

[1] Baire R. Sur les fonctions de variables réelles//Annali di mat. pura ed appl., ser.3. – 1899. – 3. – P.1–123.

[2] Piotrowski Z. Separate and joint coutinuity// Real Anal. Exch. – 1985 – 1986. – 11, N2. – P.293–322.

[3] Piotrowski Z. Separate and joint continuity.II// Real Anal. Exch. – 1989 – 1990. – 15, N1. – P.248–258.

[4] Maslyuchenko V.K., Mykhailiuk V.V., Sobchuk O.V. Studies on distinctly continuous mappings //Materials of the International Math. Conference dedicated to the memory of Hans Hahn. – Chernivtsi: Ruta, 1995. – P.192–246.

[5] Maslyuchenko V.K. Hahn spaces and the Dini problem//Mat. methods and phys.-mechan. fields. – 1998. – 41, N4. – P.39–45.

[6] Maslyuchenko V.K. The Dini problem and uniform continuity// Nauk. visnyk Chernivtsi un-tu: Zb. nauk. pr. Vyp.46. Mathematics. – Chernivtsi: Chernivtsi State University, 1999. – P.80–87.

[7] Namioka J. Separate continuity and joint continuity // Pasif. J. Math. – 1974. – 51, N2. – P.515–531.

[8] Maslyuchenko V.K. A modern version of one result of R.Baire//International conference dedicated to J.P.Schauder. Lviv, August 23-29, 1999. Nonlinear partial differential equations. Book of abstracts. – Lviv,1999. – P.139.

[9] Neubrunn T. Quasi-continuity// Real Anal.Exch. – 1988 – 1989. – 14, N3. – P.259–306.

[10] Engelking R. General topology. – M.: Mir, 1986. – 752 p.

[11] Maslyuchenko V.K., Nesterenko V.V. On the continuity of distinctly continuous mappings on curves // Mat. Studii. – 1998. – 9, No. 2. – P.201–210.

[12] Hahn H. Reelle Funktionen.1.Teil. Punkffunktionen. – Leipzig: Academische Verlagsgesellschaft M.B.H.,1932. – 416 s.