Continuous horizontally rigid functions of two variables are affine (Q1759592)

A function \(f: \mathbb R^n \to \mathbb R\) is called horizontally rigid if the graph of the function \(f(c\;\cdot)\) is isometric to the graph of \(f\) for all positive real numbers \(c\). The authors solve a system of functional equations. The result is used to the proof of the following theorem. Theorem. A continuous function \(f:\mathbb R^2 \to \mathbb R\) is horizontally rigid if and only if it is of the form \(a+bx+dy\) \((a, b, d \in \mathbb{R})\).