Tangents of $\sigma$-finite Curves and Scaled Oscillation

TL;DR

This paper proves that continuous simple curves with $\sigma$-finite length have tangents at many points and explores related properties of functions with finite lower scaled oscillation, extending the results to higher dimensions.

Contribution

It establishes the existence of tangents at many points for $\sigma$-finite length curves and applies this to functions with finite lower scaled oscillation, including higher-dimensional cases.

Findings

Curves with $\sigma$-finite length have tangents at many points.

Results extend to functions with finite lower scaled oscillation.

Study of tangent properties in higher dimensions.

Abstract

We show that every continuous simple curve with $\sigma$-finite length has a tangent at positively many points. We also apply this result to functions with finite lower scaled oscillation; and study the validity of the results in higher dimension.