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Abstract

Let $X$ be a Polish space and $Y$ be a separable metric space. For a fixed $\xi <\omega_{1}$, consider a family $f_{\alpha}\colon\, X \to Y~( \alpha<\omega_{1})$ of Baire-$\xi $ functions. Answering a question of Tomasz Natkaniec, we show that if for a function $f\colon\, X \to Y$, the set $\{ \alpha < \omega_{1}\colon\, f_{\alpha}(x) \neq f(x)\}$ is finite for every $x \in X$, then $f$ itself is necessarily Baire-$\xi$. The proof is based on a characterization of $\Sigma^{0}_{\eta}$ sets which can be interesting in its own right.