Sums of connectivity functions on Rⁿ

Abstract

A function f : Rⁿ → R is a connectivity function if the graph of its restriction f|C to any connected C ⊂ Rⁿ is connected in Rⁿ x R. The main goal of this paper is to prove that every function f : Rⁿ → R is a sum of n + 1 connectivity functions (Corollary 2.2). We will also show that if n > 1, then every function g: Rⁿ → R which is a sum of n connectivity functions is continuous on some perfect set (see Theorem 2.5) which implies that the number n + 1 in our theorem is best possible (Corollary 2.6).

To prove the above results, we establish and then apply the following theorems which are of interest on their own.

For every dense G_δ-subset G of Rⁿ there are homeomorphisms h₁...,h_n of Rⁿ such that Rⁿ=G∪h₁(G)∪...∪h_n(G) (Proposition 2.4).

For every n >1 and any connectivity function f : Rⁿ → R, if x ∈ Rⁿ and ε >0 then there exists an open set U⊂Rⁿ such that x ∈ U⊂Bⁿ(x,ε), f|bd(U) is continuous, and |(x)-f(y)| < ε for every y∈bd(U) (Proposition 2.7).