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

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 $f\upharpoonright X_n$ 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 $\mathrm{\mid }X\mathrm{\mid }<\kappa$ is $\sigma$-continuous (resp. $\bar{\sigma }$-continuous). It is clear that $\omega_1\le\mathfrak c_{\bar\sigma}\le\mathfrak c_\sigma\le\mathfrak c$\mathrm{.}Weprovethat$\mathfrak p\le\mathfrak q_0=\mathfrak c_{\bar\sigma}=\min{\mathfrak c_\sigma,\mathfrak b,\mathfrak q}\le\mathfrak c_\sigma\le\min{$\mathrm{n}\mathrm{o}\mathrm{n}$(\mathcal M),$\mathrm{n}\mathrm{o}\mathrm{n}$(\mathcal N)}$\mathrm{.}Theequality$\mathfrak c_{\bar\sigma}=\mathfrak q_0$ resolves a problem from the initial version of the paper.