  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.
