On functions that are almost continuous and perfectly everywhere surjective but not Jones. Lineability and additivity

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 col and c. This lower bound can be achieved even when omega(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 subset of C(R) of cardinality omega(1) < c such that for every g is an element of C(R) the set g \ boolean OR F has cardinality less than c. Some open problems are posed as well. (C) 2017 Elsevier B.V. All rights reserved.