THE STRUCTURE OF CONTINUOUS RIGID FUNCTIONS OF TWO VARIABLES.

A function ƒ : ℝn → ℝ is called vertically rigid if graph(cf) is isometric to graph(ƒ) for all c ≠ 0. In [1] we settled Jankovic's conjecture by showing that a continuous function ƒ : ℝ → ℝ is vertically rigid if and only if it is of the form a + bx or a + bekx (a; b; k ∈ ℝ). Now we prove that a continuous function ƒ : ℝ² → ℝ is vertically rigid if and only if, after a suitable rotation around the z-axis, ƒ (x; y) is of the form a+bx+dy, a+s(y)ekx or a+bekx +dy (a; b; d; k ∈ ℝ, k ≠ 0, s : ℝ → ℝ continuous). The problem remains open in higher dimensions. [ABSTRACT FROM AUTHOR]