Real Analysis Exchange

Continuous Rigid Functions

Christian Richter

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A function $f: {\mathbb R} \rightarrow {\mathbb R}$ is vertically [horizontally] rigid for $C \subseteq (0,\infty)$ if $graph(cf)$ [$graph(f(c\;\cdot))$] is isometric with $graph(f)$ for every $c \in C$. $f$ is vertically [horizontally] rigid if this applies to $C= (0,\infty)$. Balka and Elekes have shown that a continuous function $f$ vertically rigid for an uncountable set $C$ must be of the form $f(x)=px+q$ or $f(x)=pe^{qx}+r$, in this way confirming Jancović's conjecture saying that a continuous $f$ is vertically rigid if and only if it is of one of these forms. We prove that their theorem actually applies to every $C \subseteq (0,\infty)$ generating a dense subgroup of $((0,\infty),\cdot)$, but not to any smaller $C$. A continuous $f$ is shown to be horizontally rigid if and only if it is of the form $f(x)=px+q$. In fact, $f$ is already of that kind if it is horizontally rigid for some $C$ with $card(C \cap ((0,\infty) \setminus \{1\})) \ge 2$

Article information

Real Anal. Exchange, Volume 35, Number 2 (2009), 343-354.

First available in Project Euclid: 22 September 2010

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 39B72: Systems of functional equations and inequalities
Secondary: 26A09: Elementary functions 39B22: Equations for real functions [See also 26A51, 26B25] 51M04: Elementary problems in Euclidean geometries

vertically rigid function horizontally rigid function


Richter, Christian. Continuous Rigid Functions. Real Anal. Exchange 35 (2009), no. 2, 343--354.

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