On ∞-convex sets in spaces of scatteredly continuous functions

Given a topological space X, we study the structure of ∞-convex subsets in the space S C p ( X ) of scatteredly continuous functions on X. Our main result says that for a topological space X with countable strong fan tightness, each potentially bounded ∞-convex subset F ⊂ S C p ( X ) is weakly discontinuous in the sense that each non-empty subset A ⊂ X contains an open dense subset U ⊂ A such that each function f | U , f ∈ F , is continuous. This implies that F has network weight n w ( F ) ⩽ n w ( X ) .