2 5 Ju n 20 02 The Beckman - Quarles theorem for continuous mappings from R 2 to C

Let φ : C2×C2 → C, φ((x1, x2), (y1, y2)) = (x1 −y1) +(x2 −y2). We say that f : R → C preserves distance d ≥ 0 if for each x, y ∈ R φ(x, y) = d implies φ(f(x), f(y)) = d. We prove that if x, y ∈ R and |x − y| = (2 √ 2/3) · ( √ 3) (k, l are non-negative integers) then there exists a finite set {x, y} ⊆ Sxy ⊆ R such that each unit-distance preserving mapping from Sxy to C 2 preserves the distance between x and y. It implies that each continuous map from R to C preserving unit distance preserves all distances. The classical Beckman-Quarles theorem states that each unit-distance preserving mapping from R to R (n ≥ 2) is an isometry, see [1], [2], [5] and [8]. Author’s discrete form of this theorem ([9],[10]) states that if x, y ∈ R n (n ≥ 2) and |x − y| is an algebraic number then there exists a finite set {x, y} ⊆ Sxy ⊆ R such that each unit-distance preserving mapping from Sxy to R n preserves the distance between x and y. Mathematics Subject Classification (2000): 51M05 Let φn : C n×Cn → C, φn((x1, ..., xn), (y1, ..., yn)) = (x1−y1)+ ...+(xn− yn) . We say that f : R → C preserves distance d ≥ 0 if for each x, y ∈ R φn(x, y) = d 2 implies φn(f(x), f(y)) = d . In [11] the author proved that each continuous mapping from R to C (n ≥ 3) preserving unit distance preserves all distances. In this paper we prove it for n = 2, similarly to the case n ≥ 3 the proof is based on calculations using the Cayley-Menger determinant. Proposition 1 ([11], cf. [3], [4]). The points c1 = (z1,1, ..., z1,n), ..., cn+1 = (zn+1,1, ..., zn+1,n) ∈ C are affinely dependent if and only if their CayleyMenger determinant det   0 1 1 ... 1 1 φn(c1, c1) φn(c1, c2) ... φn(c1, cn+1) 1 φn(c2, c1) φn(c2, c2) ... φn(c2, cn+1) ... ... ... ... ... 1 φn(cn+1, c1) φn(cn+1, c2) ... φn(cn+1, cn+1)   equals 0. Proof. It follows from the equality  det   z1,1 z1,2 ... z1,n 1 z2,1 z2,2 ... z2,n 1 ... ... ... ... ... zn+1,1 zn+1,2 ... zn+1,n 1     2