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Making Borel functions look continuous. (English) Zbl 0964.03051

Let be \(g:X\to {\mathbb R}\) where \(X\) is a separable metric space. The function \(g\) is called a connectivity function if the graph of the restriction of \(g\) to any connected subset of \(X\) is a connected subset of \(X\times {\mathbb R}\); \(g\) is called an extendable connectivity function if there exists a connectivity function \(G:X\times [0,1]\to {\mathbb R}\) such that \(g(x)=G(x,0)\) for any \(x\in X\).
Let \({\mathbf B}_\alpha \) denote the set of all functions \(f:{\mathbb R}\to {\mathbb R}\) for which preimages of open sets are in the Borel class \(\Sigma ^0_{\alpha +1}\) if \(\alpha <\omega \), and in the Borel class \(\Sigma ^0_\alpha \) if \(\omega \leq \alpha <\omega _1\) (i.e. the functions of the Baire class \(\alpha \) coincide with \({\mathbf B}_\alpha \) if \(\alpha <\omega \) and with \({\mathbf B}_{\alpha +1}\) if \(\alpha \geq \omega \)). The following result is proved: For any \(\alpha \), \(1\leq \alpha <\omega _1\), there exists a function \(f\) in \({\mathbf B}_\alpha \) such that for any function \(g\) in \(\bigcup _{\gamma <\alpha}{\mathbf B_\gamma}\), \(f+g\) is an extendable connectivity function. This is a generalization of two results obtained by Natkaniec and Recław.

MSC:

03E15 Descriptive set theory
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
26A21 Classification of real functions; Baire classification of sets and functions
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