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Sums of Darboux-like functions from \(\mathbf R^n\) to \(\mathbf R^m\). (English) Zbl 0967.26009

Summary: The additivity \(\text{A}(\mathcal F)\) of a family \(\mathcal F\subseteq \mathbf R^{\mathbf R}\) is the minimum cardinality of a \(G\subseteq \mathbf R^{\mathbf R}\) with the property that \(f+G\subseteq \mathcal F\) for no \(f\in \mathbf R^{\mathbf R}\). The values of \(\text{A}\) have been calculated for many families of Darboux-like functions in \(\mathbf R^{\mathbf R}\). We extend these results to include some families of Darboux-like functions in \(\mathbf R^{\mathbf R^n}\). To do this we must define \((n,k)\)-additivity which is much more flexible than additivity.

MSC:

26B05 Continuity and differentiation questions
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
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