A function f from $$Rⁿ to R is a connectivity function if the graph of its restriction f|C to any connected subset C of $$Rⁿ is connected. The main goal of the paper is to prove that every function f:$$Rⁿ-->R is a sum of n+1 connectivity functions. We will also show that if n>1, then every function g:$$Rⁿ-->R which is a sum of n connectivity functions is continuous on some perfect set which implies that the number n+1 in our theorem is the best possible. To prove the above results, we establish and then apply the following theorem that is of interest on its own. For every dense G-delta subset G of $$Rⁿ there are homeomorphisms $h$₁,...,$h$_n of $$Rⁿ such that the sets G, $h$₁(G),...,$h$_n(G) cover $$Rⁿ.

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Last modified May 3, 1998.