Open Access
2005/2006 Vertically rigid functions.
Brandi Cain, John Clark, David Rose
Author Affiliations +
Real Anal. Exchange 31(2): 515-518 (2005/2006).


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.


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Brandi Cain. John Clark. David Rose. "Vertically rigid functions.." Real Anal. Exchange 31 (2) 515 - 518, 2005/2006.


Published: 2005/2006
First available in Project Euclid: 10 July 2007

zbMATH: 1103.54010
MathSciNet: MR2265791

Primary: ‎54C30
Secondary: 33B10 , 51M04 , 54A10

Keywords: group homomorphism , vertically rigid function

Rights: Copyright © 2005 Michigan State University Press

Vol.31 • No. 2 • 2005/2006
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