A dichotomy for Borel functions

TL;DR

This paper extends Solecki's dichotomy for Baire class 1 functions to all Borel functions, providing a simpler proof and a stronger statement about their structure, distinguishing between -continuity and inclusion of the Pawlikowski function.

Contribution

It offers a simpler proof and a broader dichotomy theorem applicable to all Borel functions, not just Baire class 1.

Findings

A dichotomy for all Borel functions is established.

The proof is simpler than Solecki's original.

The result strengthens the understanding of Borel function structure.

Abstract

The dichotomy discovered by Solecki in \cite{Sol} states that any Baire class 1 function is either $\sigma$-continuous or "includes" the Pawlikowski function $P$. The aim of this paper is to give an argument which is simpler than the original proof of Solecki and gives a stronger statement: a dichotomy for all Borel functions.