Continuity of scalar-fields characterized by smooth paths fulfilling $\|\bs(t)\|\|\bs'(t)\| < +\infty$

TL;DR

This paper characterizes the continuity of scalar functions at the origin using a restricted class of smooth paths with bounded product of norm and derivative, and constructs paths matching given sequences with specific properties.

Contribution

It introduces a smaller class of paths sufficient for characterizing continuity and demonstrates the existence of paths matching prescribed sequences with certain geometric constraints.

Findings

A smaller class of smooth paths suffices for continuity characterization.

Existence of paths matching sequences with prescribed directions and magnitudes.

Path construction under specific sequence and geometric conditions.

Abstract

A function $f$ from a subset of $\R^n$ to $\R$ is continuous at the origin, if and only if $\lim_{t\to 0+} f(\bs(t))=f(\bnull)$ for all continuous paths $\bs$ with $\lim_{t\to 0+} \bs(t)=\bnull$. The continuity of $f$ can, however, be characterized by a much smaller class of paths. We show that the class of all paths fulfilling $\lim_{t\to 0+} \bs(t)=\bnull$, $\bs\in [\cC^\infty(]0,a[)]^n$, and $\sup_{t\in\,]0,a[}\|\bs(t)\|\|\bs'(t)\| < +\infty$ is sufficient. Further, given any sequences $(\bx_k)_{k\in\N}$ and $(\by_k)_{k\in\N}$ in $\R^n\setminus\{\bnull\}$, such that $\lim_{k\to+\infty}\bx_k=\bnull$, $\bx_k\cdot\by_k \ge 0$, and $\|\by_k\|=1$ for all $k\in\N$, we show that there exist a path of this class, such that $\bs(\|\bx_k\|)=\bx_k$ and $\bs'(\|\bx_k\|)=\by_k$ for an infinite number of $k\in\N$.