## Continuous rigid functions.(English)Zbl 1225.39026

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.

### MSC:

 39B72 Systems of functional equations and inequalities 26A27 Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives 39B22 Functional equations for real functions

### Keywords:

rigid functions; functional equations
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