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 $[a,b]$,

The total variation $V_{\gamma}$ of $\gamma$ is defined by

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

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.