Suppose that $(X,d)$ is a metric space. Let $f$ be a curve, so that $f: [0,1] \to X$ is a continuous function, and let $0=t_0 < t_1 < \cdots < t_n=1$ and $x_i = f(t_i)$ for $0 \le i \le n$ The set $\{x_0, x_1, \ldots , x_n\}$ is called a partition of the curve. The length of the curve is defined to be the supremum over all partitions of the quantity $\sum_{i=1}^n d(x_i , x_{i-1})$