It is investigated necessary and sufficient conditions on such system  $\mathcal{A}$ of subsets of a topological space $X$ that for every function  $f: X → \mathbb{R}$  the continuity of all restrictions $f|_A,$  where $A ∈ \mathcal{A}$, on $A$ (or in some point $x_0 ∈ X$) implies the continuity $f$ on $X$ (or in $x_0$). 

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