When does SC(X)=RXSC(X)=RX hold?

A map f:X→Yf:X→Y between topological spaces is called scatteredly continuous if for each non-empty subspace A⊂XA⊂X the restriction f|Af|A has a point of continuity. By SC(X)SC(X) we denote the set of all scatteredly continuous maps from XX to the space of real numbers RR. We consider the following problem: What conditions must satisfy space XX so that SC(X)=RXSC(X)=RX?