LAUSR

Abstract We show that the class of functions that are perfectly everywhere surjective and almost continuous in the sense of Stallings but are not Jones functions is c + -lineable. Moreover, it is consistent that this class is 2 c -lineable, as this holds when 2 c = c . We also prove that the additivity number for this class is between ω 1 and c . This lower bound can be achieved even when ω 1 c , as it is implied by the Covering Property Axiom CPA. The main step in this proof is the following theorem, which is of independent interest: CPA implies that there exists a family F ⊂ C ( R ) of cardinality ω 1 c such that for every g ∈ C ( R ) the set g ∖ ⋃ F has cardinality less than c . Some open problems are posed as well.