Let $\gamma:[a,b]\rightarrow X$ be a function mapping an interval $[a,b]$ to a metric space $(X,d)$ . We say that $\gamma$ is of bounded variation if there is a constant $M$ such that, for each partition $P=\{a=t_0<t_1<\cdots < t_n=b\}$ of $[a,b]$ , $$ v(\gamma, P)= \sum_{k=1}^n d(\gamma(t_k),\gamma(t_{k-1})) \leq M. $$ The total variation $V_\gamma$ of $\gamma$ is defined by $$ V_\gamma = \sup\{v(\gamma,P):\textnormal{$P$ is a partition of $[a,b]$}\} $$

It can be shown that, if $X$ is either $\mathbb{R}$ or $\mathbb{C}$ , every continuously differentiable (or piecewise continuously differentiable) function $\gamma:[a,b]\rightarrow X$ is of bounded variation, and $$ V_\gamma = \int_a^b |\gamma'(t)|dt $$ Also, if $\gamma$ is of bounded variation and $f:[a,b]\rightarrow X$ is continuous, then the Riemann-Stieltjes integral $\int_a^b fd\gamma$ is finite.

If $\gamma$ is also continuous, it is said to be a rectifiable path, and $V(\gamma)$ is the length of its trace.

If $X=\mathbb{R}$ , it can be shown that $\gamma$ is of bounded variation if and only if it is the difference of two monotonic functions.