# $\sigma$-Continuous functions and related cardinal characteristics of the continuum

## TL;DR

This paper investigates $\sigma$-continuous functions and introduces cardinal characteristics of the continuum, establishing relationships and equalities among them, which resolve previous open problems in the field.

## Contribution

The paper defines new cardinal invariants related to $\sigma$-continuity and proves their exact relationships, resolving an open problem about these characteristics.

## Key findings

- Established that $rak p \, extless\, rak q_0=\mathfrak c_{\bar\sigma}$
- Proved that $rak c_{\bar\sigma} = \min\{rak c_{\sigma}, \frak b, \frak q\}$
- Showed inequalities linking these invariants to classical cardinal characteristics

## Abstract

A function $f:X\to Y$ between topological spaces is called $\sigma$-$continuous$ (resp. $\bar\sigma$-$continuous$) if there exists a (closed) cover $\{X_n\}_{n\in\omega}$ of $X$ such that for every $n\in\omega$ the restriction $f{\restriction}X_n$ is continuous. By $\mathfrak c_\sigma$ (resp. $\mathfrak c_{\bar\sigma}$) we denote the largest cardinal $\kappa\le\mathfrak c$ such that every function $f:X\to\mathbb R$ defined on a subset $X\subset\mathbb R$ of cardinality $|X|<\kappa$ is $\sigma$-continuous (resp. $\bar\sigma$-continuous). It is clear that $\omega_1\le\mathfrak c_{\bar\sigma}\le\mathfrak c_\sigma\le\mathfrak c$. We prove that $\mathfrak p\le\mathfrak q_0=\mathfrak c_{\bar\sigma}=\min\{\mathfrak c_\sigma,\mathfrak b,\mathfrak q\}\le\mathfrak c_\sigma\le\min\{\mathrm{non}(\mathcal M),\mathrm{non}(\mathcal N)\}$. The equality $\mathfrak c_{\bar\sigma}=\mathfrak q_0$ resolves a problem from the initial version of the paper.