A function f from
$$**R**^{n} to **R** is a connectivity function if
the graph of
its restriction f|C to any connected subset C of
$$**R**^{n} is connected.
The main goal of the paper is to prove that every function
f:$$**R**^{n}-->**R**
is a sum of n+1 connectivity functions.
We will also show that if n>1, then every function
g:$$**R**^{n}-->**R** which
is a sum of n connectivity functions is continuous on some perfect
set 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
$$**R**^{n} there are homeomorphisms
$h$_{1},...,$h$_{n}
of $$**R**^{n} such that the sets G,
$h$_{1}(G),...,$h$_{n}(G)
cover $$**R**^{n}.

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**Last modified May 3, 1998.**