Real Analysis Exchange

The Structure of Continuous Rigid Functions of Two Variables

Richárd Balka and Márton Elekes

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Abstract

A function $f : \mathbb{R}^n \to \mathbb{R}$ is called vertically rigid if $graph(cf)$ is isometric to $graph (f)$ for all $c \neq 0$. In [1] settled Janković's conjecture by showing that a continuous function $f : \mathbb{R}\to \mathbb{R}$ is vertically rigid if and only if it is of the form $a+bx$ or $a+be^{kx}$ ($a,b,k \in )$ prove that a continuous function $f :\mathbb{R}^2 \to \mathbb{R}$ is vertically rigid if and only if, after a suitable rotation around the $z$-axis, $f(x,y)$ is of the form $a + bx + dy$, $a + s(y)e^{kx}$ or $a + be^{kx} + dy$ ($a,b,d,k \in \mathbb{R}$, $k \neq 0$, $s : \mathbb{R} \to \mathbb{R}$ continuous). The problem remains open in higher dimensions.

Article information

Source
Real Anal. Exchange, Volume 35, Number 1 (2009), 139-156.

Dates
First available in Project Euclid: 27 April 2010

Permanent link to this document
https://projecteuclid.org/euclid.rae/1272376228

Mathematical Reviews number (MathSciNet)
MR2657292

Zentralblatt MATH identifier
1204.26021

Subjects
Primary: 26A99: None of the above, but in this section
Secondary: 39B22: Equations for real functions [See also 26A51, 26B25] 39B52: Equations for functions with more general domains and/or ranges 39B72: Systems of functional equations and inequalities 51M99: None of the above, but in this section

Keywords
rigid functional equation transformation exponential

Citation

Balka, Richárd; Elekes, Márton. The Structure of Continuous Rigid Functions of Two Variables. Real Anal. Exchange 35 (2009), no. 1, 139--156. https://projecteuclid.org/euclid.rae/1272376228


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References

  • R. Balka, M. Elekes, The structure of rigid functions, J. Math. Anal. Appl., 345(2) (2008), 880–888.
  • B. Cain, J. Clark, D. Rose, Vertically rigid functions, Real Anal. Exchange, 31(2) (2005/2006), 515–518.
  • C. Richter, Continuous rigid functions, preprint. (http://www.minet.uni-jena.de/Math-Net/reports/shadows//08-02report.html)