Cardinal invariants concerning functions whose sum is almost continuous

TL;DR

This paper investigates the cardinal invariants related to functions whose sum is almost continuous, establishing bounds on their cofinality and showing their possible values within certain cardinal ranges.

Contribution

It proves that the cofinality of the invariant AA exceeds the continuum and that AA can take any regular cardinal value between c^+ and 2^c, also relating AA to Darboux functions.

Findings

Cofinality of AA is greater than the continuum.

AA can equal any regular cardinal between c^+ and 2^c.

AA equals AD, solving a related problem.

Abstract

A function f from reals to reals (f:R->R) is almost continuous (in the sense of Stallings) iff every open set in the plane which contains the graph of f contains the graph of a continuous function. Natkaniec showed that for any family F of continuum many real functions there exists g:R->R such that f+g is almost continuous for every f in F. Let AA be the smallest cardinality of a family F of real functions for which there is no g:R->R with the property that f+g is almost continuous for every f in F. Thus Natkaniec showed that AA is strictly greater than the continuum. He asked if anything more could be said. We show that the cofinality of AA is greater than the continuum, c. Moreover, we show that it is pretty much all that can be said about AA in ZFC, by showing that AA can be equal to any regular cardinal between c^+ and 2^c (with 2^c arbitrarily large). We also show that AA = AD where AD is defined similarly to AA but for the class of Darboux functions. This solves another problem of Maliszewski and Natkaniec.