## Sums of bounded Darboux functions.(English)Zbl 0830.26002

The following results are proved: 1. For a family $$\mathcal F$$ of bounded functions with a common bound there is a bounded function $$h$$ such that $$h+ f$$ is Darboux for all $$f\in {\mathcal F}$$ if and only if the cardinality of $$\mathcal F$$ is less than the cofinality of the continuum. Moreover, if the cardinality of $$\mathcal F$$ is less than the additivity of the ideal of null subsets of $$\mathbb{R}$$ (of meager sets, respectively) and all functions $$f\in {\mathcal F}$$ are Lebesgue measurable (have the Baire property) then we can require that $$h$$ is measurable (has the Baire property). 2. For a countable family $$\mathcal F$$ of Baire $$\alpha$$ functions with a common bound there exists a bounded Baire 2 function $$h$$ such that $$f+ h$$ is Darboux for each $$f\in {\mathcal F}$$. 3. For a finite family $$\mathcal F$$ of Baire one functions with a common bound there exists a bounded Baire one function $$h$$ such that $$f+ h$$ is Darboux for each $$f\in {\mathcal F}$$. The analogous theorem does not hold for infinite $$\mathcal F$$. 4. Suppose $$\mathcal F$$ is one of the following families of real functions: all functions, all Lebesgue measurable functions, all functions with the Baire property, Baire class $$\alpha$$. Then each bounded $$f\in {\mathcal F}$$ can be expressed as the sum of two bounded Darboux functions in the class $$\mathcal F$$.

### MSC:

 26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable 26A21 Classification of real functions; Baire classification of sets and functions 54C08 Weak and generalized continuity