## 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

### Keywords:

Baire classes; Borel functions; Darboux functions