Continuous rigid functions

Preprint series: 08-02, Reports on Analysis

C. Richter

Preprint series: , Reports on Analysis

MSC:
39B72 Systems of functional equations and inequalities
26A09 Elementary functions
51M04 Elementary problems in Euclidean geometries

Abstract: 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 one
of the forms, $f(x)=px+q$ or $f(x)=pe^{qx}+r$, this way
confirming Jancovi{\'c}'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 $card(C \cap ((0,\infty) \setminus \{1\})) \ge 2$.

Keywords: vertically rigid function, horizontally rigid function

Notes: submitted