by **
C. Richter**

**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
- 39B22 Equations for real functions [See also 26A51, 26B25]
- 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

**Upload:** 2008-02-08

**Update:** 2008

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