Title: Cardinal invariants concerning functions
whose sum is almost continuous.
Authors: Krzysztof Ciesielski and Arnold W. Miller
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.