Real Analysis Exchange

Vertically rigid functions.

Brandi Cain, John Clark, and David Rose

Full-text: Open access


A function $f\colon{\mathbb{R}}\to{\mathbb{R}}$ is said to be vertically rigid provided its graph $G(f)=\{{\langle} x,f(x){\rangle}\colon x\in{ \mathbb{R}}\}$ is isometric with the graph of the function $kf$ for every non-zero $k\in{\mathbb{R}}$. We show that the group homomorphisms $f$ from ${ \langle}{\mathbb{R}},+{\rangle}$ into ${\langle}{\mathbb{R}}^+,\cdot{\rangle} $ is vertically rigid if and only if it is an epimorphism. Some other examples of vertically rigid functions will also be given. A problem of characterizing all vertically rigid functions remains open.

Article information

Real Anal. Exchange, Volume 31, Number 2 (2005), 515-518.

First available in Project Euclid: 10 July 2007

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 54C30: Real-valued functions [See also 26-XX]
Secondary: 51M04: Elementary problems in Euclidean geometries 33B10: Exponential and trigonometric functions 54A10: Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)

vertically rigid function group homomorphism


Cain, Brandi; Clark, John; Rose, David. Vertically rigid functions. Real Anal. Exchange 31 (2005), no. 2, 515--518.

Export citation


  • J. Aczél, Lectures on Functional Equations and their Applications, Academic Press, New York and London, 1966.
  • A. L. Cauchy, Cours d'Analyse de l'Ecole Polytechnique, 1. Analyse Algébrique, V., Paris, 1821, (Oeuvres (2) 3, Paris, 1897).
  • K. Ciesielski, Set Theory for the Working Mathematician, Cambridge University Press, (1997).
  • C. F. Gauss, Theoria motus corporum coelestium, Hamburg, 1809, (Werke VII, Leipzig, 1906; pp. 240–244).
  • G. Hamel, Einer basis aller zahlen und die unstetigen l ösungen der funktionalgleichung $f(x+y)=f(x)+f(y),$ Math. Ann., 60 (1905), 459– 462.
  • M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, Uniwersytet Śląski, (1985).
  • M. Laczkovich, Conjecture and Proof, The Mathematical Association of America, (2001).
  • A. M. Legendre, Eléments de géometrie, Paris, 1791, Note II.