Cardinal invariants concerning bounded families of extendable and almost continuous functions

In this paper we introduce and examine a cardinal invariant $\operatorname{A}_{{b}}$ closely connected to the addition of bounded functions from $\mathbb{R}$ to $\mathbb{R}$. It is analogous to the invariant $\operatorname{A}$ defined earlier for arbitrary functions by T. Natkaniec. In particular, it is proved that each bounded function can be written as the sum of two bounded almost continuous functions, and an example is given that there is a bounded function which cannot be expressed as the sum of two bounded extendable functions.