The Beckman-Quarles theorem for continuous mappings from R^2 to C^2

Abstract

Let \phi((x_1,x_2),(y_1,y_2))=(x_1-y_1)^2+(x_2-y_2)^2. We say that f:R^2 -> C^2 preserves distance d>=0 if for each x,y \in R^2 \phi(x,y)=d^2 implies \phi(f(x),f(y))=d^2. We prove that if x,y \in R^2 and |x-y|=(2\sqrt{2}/3)^k \cdot (\sqrt{3})^l (k,l are non-negative integers) then there exists a finite set {x,y} \subseteq S(x,y) \subseteq R^2 such that each unit-distance preserving mapping from S(x,y) to C^2 preserves the distance between x and y. It implies that each continuous map from R^2 to C^2 preserving unit distance preserves all distances.