$\sigma$-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 $f{\restriction}X_n$ 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.