It is proved a new generalization of the Namioka theorem. In particular, it implies the following result. Let $X_1, ..., X_d$ be strongly countable complete spaces, $Y$ be a metrizable space and $γ: \prod_{i=1}^{d-1} X_i → X_d$ be compactvalued upper quasicontinuous mapping. Then for any separately continuous function $f = \prod_{i=1}^d X_i → Y$ there exists dense $G_δ$ -subset  $A ⊆ \prod_{i=1}^{d-1} X_i$ such that f is continuous in every point of  $\{x\} × γ(x)$ for any $x ∈ A$. Besides, it is obtained some new results on automatically continuity of group operations and actions.

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