CARDINAL INVARIANTS CONCERNING FUNCTIONS WHOSE SUM IS ALMOST CONTINUOUS

Abstract

Let $\mathcal A$ stand for the class of all almost continuous functions from $ℝ$ to $ℝ$ and let $\mathrm A(\mathcal A)$ be the smallest cardinality of a family $F ⊆ ℝ^ℝ$ for which there is no $g: ℝ → ℝ$ with the property that $f + g ϵ \mathcal A$ for all $f ϵ F$. We define cardinal number $A(\mathcal D)$ for the class $\mathcal D$ of all real functions with the Darboux property similarly. It is known, that $c < \mathrm A(\mathcal A) ≤ 2^c$ [11]. We will generalize this result by showing that the cofinality of $\mathrm A(\mathcal A)$ is greater that $c$. Moreover, we will show that it is pretty much all that can be said about $\mathrm A(\mathcal A)$ in ZFC, by showing that $\mathrm A(\mathcal A)$ can be equal to any regular cardinal between $c⁺$ and $2^c$ and that it can be equal to $2^c$ independently of the cofinality of $2^c$. This solves a problem of T. Natkaniec [11, Problem 6.1, p. 495].

We will also show that $\mathrm A(\mathcal D) = \mathrm A(\mathcal A)$ and give a combinatorial characterization of this number. This solves another problem of Natkaniec. (Private communication.)