On ∞-convex sets in spaces of scatteredly continuous functions

Given a topological space X, we study the structure of ∞-convex subsets in the space SCp(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⊂SCp(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 nw(F)⩽nw(X).