The structure of rigid functions

A function f : R→R is called vertically rigid if graph(cf ) is isometric to graph( f ) for all c = 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 +bekx (a, b,k ∈ R). We answer the 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.