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 be the supremum over all partitions of the quantity ${\sum }_{i=1}^{n}d(x_i,x_{i-1})$.

Title	length of curve in a metric space
Canonical name	LengthOfCurveInAMetricSpace
Date of creation	2013-03-22 16:50:27
Last modified on	2013-03-22 16:50:27
Owner	Mathprof (13753)
Last modified by	Mathprof (13753)
Numerical id	8
Author	Mathprof (13753)
Entry type	Definition
Classification	msc 26B15
Defines	length of a curve