Continuous rigid functions (Q541378)

A function \(f:{\mathbb R}\to{\mathbb R}\) is called vertically rigid for \(C\subseteq (0,\infty)\) if graph\(\left(cf\right)\) and graph\(\left(f\right)\) are isometric for every \(c\in C\). It is known that if \(C\) is ``large enough'' (say, uncountable), then any continuous, vertically rigid function for \(C\) must be of the form either \(f(x)=px+q\) or \(f(x)=pe^{qx}+r\). In the present paper, the author improves this result by showing the same conclusion for any \(C\) generating a dense subgroup of \(\left((0,\infty),\cdot\right)\). He also shows that this is best possible, i.e., that the conclusion does not hold for any smaller \(C\). A function \(f:{\mathbb R}\to{\mathbb R}\) is called horizontally rigid for \(C\subseteq (0,\infty)\) if graph\(\left(f(c\,\cdot)\right)\) and graph\(\left(f\right)\) are isometric for every \(c\in C\). In this case, the author proves that if \(C\) contains at least two different values \(c_1,c_2 \neq 1\), then any continuous, horizontally rigid function for \(C\) must be of the form \(f(x)=px+q\), which is again best possible.