On normality of spaces of scatteredly continuous maps

A map $fcolon Xrightarrow Y$ between topological spaces is called scatteredly continuous if for each non-empty subspace $Asubset X$ the restriction $f|_{A}$ has a point of continuity. By $SC_p (X)$ we denote the space of all scatteredly continuous real-valued functions on $X$ endowed with the topology of pointwise convergence. In this paper we focus on the normality of the space $SC_p(X)$. Particularly, it is proved that if the function space $SC_p(X)$ is normal, then all compact and all scattered subspaces of $X$ are countable.