When does SC(X)=RX hold?

Abstract

A map $f:X\to Y$ between topological spaces is called scatteredly continuous if for each non-empty subspace $A\subset X$ the restriction $f{|}_A$ has a point of continuity. By $SC\left(X\right)$ we denote the set of all scatteredly continuous maps from $X$ to the space of real numbers $R$. We consider the following problem: What conditions must satisfy space $X$ so that $SC\left(X\right)=R^X$?