Let $(X,d)$ be a [metric space](/page/Metric%20Space), let $\gamma:[0,1]\to X$ be a continuous path, and let $\phi:[0,1]\to[0,1]$ be a continuous, nondecreasing, surjective map satisfying $\phi(0)=0$ and $\phi(1)=1$. Let $\mathcal{P}([0,1])$ denote the set of finite partitions $P=(r_0,\dots,r_m)$ of $[0,1]$ satisfying $0=r_0<r_1<\cdots<r_m=1$. Define the length functional $L:X^{[0,1]}\to[0,\infty]$ by

\begin{align*} L(\alpha):=\sup\left\{\sum_{i=1}^{m} d(\alpha(r_i),\alpha(r_{i-1})):P=(r_0,\dots,r_m)\in\mathcal{P}([0,1])\right\} \end{align*}

for every map $\alpha:[0,1]\to X$. Then

\begin{align*} L(\gamma\circ\phi)=L(\gamma). \end{align*}