We prove that if a Δ¹₁ function f with Σ¹₁ domain X is σ-continuous then one can find a Δ¹₁ covering ${\left(Aₙ\right)}_{n\in \omega }$ of X such that $f_{|Aₙ}$ is continuous for all n. This is an effective version of a recent result by Pawlikowski and Sabok, generalizing an earlier result of Solecki.

It is shown that $n$ times Peano differentiable functions defined on a closed subset of ${ℝ}^m$ and satisfying a certain condition on that set can be extended to $n$ times Peano differentiable functions defined on ${ℝ}^m$ if and only if the $n$th order Peano derivatives are Baire class one functions.

A classical theorem of Kuratowski says that every Baire one function on a $G_{\delta }$ subspace of a Polish (= separable completely metrizable) space X can be extended to a Baire one function on X. Kechris and Louveau introduced a finer gradation of Baire one functions into small Baire classes. A Baire one function f is assigned into a class in this hierarchy depending on its oscillation index β(f). We prove a refinement of Kuratowski’s theorem: if Y is a subspace of a metric space X and f is a real-valued...

We prove an abstract version of the Kuratowski extension theorem for Borel measurable maps of a given class. It enables us to deduce and improve its nonseparable version due to Hansell. We also study the ranges of not necessarily injective Borel bimeasurable maps f and show that some control on the relative classes of preimages and images of Borel sets under f enables one to get a bound on the absolute class of the range of f. This seems to be of some interest even within separable spaces.