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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
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