Extending connectivity functions on Rn

A function f :Rn→R is a connectivity function if for every connected subset C of Rn the graph of the restriction f↾C is a connected subset of Rn+1, and f is an extendable connectivity function if f can be extended to a connectivity function g :Rn+1→R with Rn embedded into Rn+1 as Rn×{0}. There exists a connectivity function f :R→R that is not extendable. We prove that for n⩾2 every connectivity function f :Rn→R is extendable.