  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.
