Let $(X,d)$ be a [complete metric space](/page/Complete%20Metric%20Space), and let $\gamma:[0,1]\to X$ be an absolutely continuous curve. For $t\in(0,1)$, define the metric derivative, when the limit exists, by
where $h$ is restricted by $t+h\in[0,1]$. Then $|\gamma'|(t)$ exists for $\mathcal L^1$-a.e. $t\in(0,1)$, the resulting function $|\gamma'|$ belongs to $L^1((0,1),\mathcal L^1)$, and