Decomposing symmetrically continuous and Sierpinski-Zygmund functions into continuous functions

In this paper we will investigate the smallest cardinal number $\kappa$ such that for any symmetrically continuous function $f\colon\mathbb{R}\to\mathbb{R}$ there is a partition $\{X_\xi\colon\xi<\kappa\}$ of $\mathbb{R}$ such that every restriction $f\restriction X_\xi\colon X_\xi\to\mathbb{R}$ is continuous. The similar numbers for the classes of Sierpinski-Zygmund functions and all functions from $\mathbb{R}$ to $\mathbb{R}$ are also investigated and it is proved that all these numbers are equal. We also show that $\mathrm{cf}(\mathfrak{c})\leq\kappa\leq\mathfrak{c}$ and that it is consistent with ZFC that each of these inequalities is strict.