The Beckman-Quarles theorem for continuous mappings from R^n to C^n

Abstract

Let ϕ((x_1,...,x_n),(y_1,...,y_n))=(x_1-y_1)^2+...+(x_n-y_n)^2. We say that f:R^n -> C^n preserves distance d>=0 if for each x,y \in R^n ϕ(x,y)=d^2 implies ϕ(f(x),f(y))=d^2. We prove that if x,y \in R^n (n>=3) and |x-y|=(\sqrt{2+2/n})^k \cdot (2/n)^l (k,l are non-negative integers) then there exists a finite set {x,y} \subseteq S(x,y) \subseteq R^n such that each unit-distance preserving mapping from S(x,y) to C^n preserves the distance between x and y. It implies that each continuous map from R^n to C^n (n>=3) preserving unit distance preserves all distances.