Sums of Darboux-Like Functions from ℝn to ℝm

Abstract

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