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Vertically rigid functions. (English) Zbl 1103.54010

Summary: A function \(f:\mathbb{R}\to\mathbb{R}\) is said to be vertically rigid provided its graph \(G(f)=\{\langle x,f(x)\rangle:x\in\mathbb{R}\}\) is isometric to the graph of the function \(kf\) for every non-zero \(k\in\mathbb{R}\). We show that a group homomorphism \(f\) 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. The problem of characterizing all vertically rigid functions remains open.

MSC:

54C30 Real-valued functions in general topology
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)
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