It is proved that for a Baire space $X,$ a metrizable compact $Y,$ a metrizable space $Z$ and a compact-valued mapping $F: X × Y ⊸ Z$ which is lower semi-continuous with respect the first variable and continuous with respect to the second variable there exists a dense $G_δ$-set  $A ⊆ X$ such that the restriction $F|_{A × Y}$ is continuous. It is constructed an example of a mapping $F : [0,1]^2 ⊸ [0,1]$  which is lower semi-continuous with respect the first variable, continuous with respect to the second variable and jointly discontinuous at every point of the set $[0,1] × \{0\}$.

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