length of curve in a metric space

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 and$x_i=f(t_i)$ for $0\le i\le n$.The set $\{x_0,x_1,\mathrm{\dots },x_n\}$is called a partition of the curve.The of the curve is defined to bethe supremum over all partitions of the quantity ${\sum }_{i=1}^{n}d(x_i,x_{i-1})$.