Let $(X,d)$ be a [metric space](/page/Metric%20Space), and let $\gamma:[0,1]\to X$ be a path, meaning that $\gamma$ is continuous with respect to the [Euclidean metric](/page/Euclidean%20Metric) on $[0,1]$ and the metric $d$ on $X$. Then $\gamma$ is uniformly continuous: for every $\varepsilon>0$ there exists $\delta>0$ such that for all $s,t\in[0,1]$,

\begin{align*} |s-t|<\delta \implies d(\gamma(s),\gamma(t))<\varepsilon. \end{align*}