Continuity of scalar-fields characterized by smooth paths fulfilling $\mathrm{\parallel }\text{bs}(t)\mathrm{\parallel }\mathrm{\parallel }{\text{bs}}^{\mathrm{\prime }}(t)\mathrm{\parallel }<+\mathrm{\infty }$

A function $f$ from a subset of ${\mathbb{R}}^n$ to $\mathbb{R}$ is continuous at the origin, if and only if ${\lim }_{t\to 0+}f(\text{bs}(t))=f(\text{bnull})$ for all continuous paths $\text{bs}$ with ${\lim }_{t\to 0+}\text{bs}(t)=\text{bnull}$. The continuity of $f$ can, however, be characterized by a much smaller class of paths. We show that the class of all paths fulfilling ${\lim }_{t\to 0+}\text{bs}(t)=\text{bnull}$, $\bs\in [\cC^\infty(]0,a[)]^n$,and$\sup_{t\in,]0,a[}|\bs(t)||\bs’(t)| < +\infty$issufficient\mathrm{.}Further,givenanysequences$(\bx_k){k\in\N}$and$(\by_k){k\in\N}$in$\R^n\setminus{\bnull}$,suchthat$\lim_{k\to+\infty}\bx_k=\bnull$,$\bx_k\cdot\by_k \ge 0$,and$|\by_k|=1$forall$k\in\N$,weshowthatthereexistapathofthisclass,suchthat$\bs(|\bx_k|)=\bx_k$and$\bs’(|\bx_k|)=\by_k$foraninfinitenumberof$k\in\N$.