Real Analysis Exchange

Vertically rigid functions.

Brandi Cain, John Clark, and David Rose

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Abstract

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

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

Dates
First available in Project Euclid: 10 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.rae/1184104042

Mathematical Reviews number (MathSciNet)
MR2265791

Zentralblatt MATH identifier
1103.54010

Subjects
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)

Keywords
vertically rigid function group homomorphism

Citation

Cain, Brandi; Clark, John; Rose, David. Vertically rigid functions. Real Anal. Exchange 31 (2005), no. 2, 515--518. https://projecteuclid.org/euclid.rae/1184104042


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