Sums of Connectivity Functions on R n

A function f : R n → R is a connectivity function if the graph of its restriction f | C to any connected C ⊂ R n is connected in R n × R. The main goal of this paper is to prove that every function f : R n → R is a sum of n + 1 connectivity functions (Corollary 2.2). We will also show that if n > 1, then every function g : R n → 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). Toprove 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 R n there are homeomorphisms h 1 , …, h n of R n such that R n = G ∪ h 1 ( G ) ∪ … ∪ h n ( G ) (Proposition 2.4). For every n > 1 and any connectivity function f : R n → R, if x ∈ R n and ε > 0 then there exists an open set U ⊂ R n such that x ∈ U ⊂ B n ( x , ε), f |bd( U ) is continuous, and |( x ) − f ( y )| < ε for every y ∈ bd( U ) (Proposition 2.7). 1991 Mathematics Subject Classification : 26B40, 54C30, 54F45.