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arXiv · 2410.12962

Graphs of continuous but non-affine functions are never self-similar

Carlos Gustavo Moreira·Jinghua Xi·Yiwei Zhang
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Abstract

Bandt and Kravchenko \cite{BandtKravchenko2010} proved that if a self-similar set spans $\R^m$, then there is no tangent hyperplane at any point of the set. In particular, this indicates that a smooth planar curve is self-similar if and only if it is a straight line. When restricting curves to graphs of continuous functions, we can show that the graph of a continuous function is self-similar if and only if the graph is a straight line, i.e., the underlying function is affine.

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Carlos Gustavo Moreira, Jinghua Xi, Yiwei Zhang. 2024-10-16. Graphs of continuous but non-affine functions are never self-similar. https://arxiv.org/abs/2410.12962

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