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We prove that  a compact-valued multifunction  F :  X  ×  Y  →  Z , where  X  is a Baire space and  Y ,  Z  are separable metrizable spaces, is quasi-continuous if and only if  F  is horizontally quasi-continuous and there exists an residual subset  M  of  X  such that for any  x  ∈  M  the multifunction  F   x   =  F ( x , ·) is quasi-continuous on  Y .

Characterization of quasi-continuity of multifunctions of two variables