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 $2c$ (with $2c$ 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.

**Requires rae.cls file**
and amstex.sty