The structure of rigid functions

TL;DR

This paper characterizes continuous vertically rigid functions as affine or exponential functions, proves a conjecture, and explores measurable and non-regular cases, also introducing horizontally rigid functions.

Contribution

It proves Janković's conjecture for continuous functions, characterizes all such functions, and extends the analysis to measurable and non-regular functions, also defining horizontally rigid functions.

Findings

Continuous vertically rigid functions are exactly affine or exponential functions.

Existence of Borel measurable vertically rigid functions not of the standard form.

A structure theorem for horizontally rigid functions without regularity assumptions.

Abstract

A function $f:\RR \to \RR$ is called \emph{vertically rigid} if $graph(cf)$ is isometric to $graph (f)$ for all $c \neq 0$. We prove Jankovi\'c's conjecture by showing that a continuous function is vertically rigid if and only if it is of the form $a+bx$ or $a+be^{kx}$ ($a,b,k \in \RR$). We answer a question of Cain, Clark and Rose by showing that there exists a Borel measurable vertically rigid function which is not of the above form. We discuss the Lebesgue and Baire measurable case, consider functions bounded on some interval and functions with at least one point of continuity. We also introduce horizontally rigid functions, and show that a certain structure theorem can be proved without assuming any regularity.