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

TL;DR

This paper proves that the graphs of continuous, non-affine functions cannot be self-similar, extending previous results about self-similarity and tangent hyperplanes to a broader class of functions.

Contribution

It establishes that only affine functions have self-similar graphs among continuous functions, generalizing prior work on self-similarity and tangent hyperplanes.

Findings

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

Self-similar graphs of continuous functions must be straight lines.

The result extends the understanding of self-similarity in geometric function graphs.

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.