COMPOSITIONS OF TWO ADDITIVE ALMOST CONTINUOUS FUNCTIONS
Banaszewski, Ciesielski · Real Analysis Exchange · 1997
In the paper we prove that an additive Darboux function f : R → R can be expressed as a composition of two additive almost continuous (connectivity) functions if and only if either f is almost continuous (connectivity) function or dim(ker(f )) = 1.We also show that for every cardinal number λ ≤ 2 ω there exists an additive almost continuous functions with dim(ker(f )) = λ.A question whether every Darboux function f : R → R can be expressed as a composition of two almost continuous functions (see [?] or [?]) remains open. Definitions and NotationOur terminology and notation is standard.In particular, functions will be identified with their graphs, and for a subset A of R × R (possibly, but not necessarily, a graph of a function) we will write dom (A) and rng (A) to denote the x-projection (the domain) and the y-projection (the range) of A, respectively.The cardinality of a set A will be denoted by card (A).Cardinals will be identified with the initial ordinals.The cardinality of the set R of real numbers, the continuum, will be denoted by 2 ω .Throughout the paper we will consider R as a linear space over the field Q of rational numbers.A linear basis of this space will be referred to as a Hamel basis.It is evident that the cardinality of every Hamel basis is equal to 2 ω .