A dichotomy property for the graphs of monomials

We prove that the graph of a discontinuous $n$-monomial function $f:\mathbb{R}\to\mathbb{R}$ is either connected or totally disconnected. Furthermore, the discontinuous monomial functions with connected graph are characterized as those satisfying a certain big graph property. Finally, the connectedn...

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Bibliographic Details
Main Authors Almira, J. M, Boros, Z
Format Journal Article
LanguageEnglish
Published 30.10.2014
Subjects
Mathematics - Classical Analysis and ODEs
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Summary:We prove that the graph of a discontinuous $n$-monomial function $f:\mathbb{R}\to\mathbb{R}$ is either connected or totally disconnected. Furthermore, the discontinuous monomial functions with connected graph are characterized as those satisfying a certain big graph property. Finally, the connectedness properties of the graphs of additive functions $f:\mathbb{R}^d\to\mathbb{R}$ are studied.
DOI:10.48550/arxiv.1410.8468