Total variation

Builds on Partition of an interval Supremum

Definition

Let $a < b$ be real numbers and let $f \colon [a, b] \to \R$ be a function. The total variation of $f$ on $[a, b]$ is $V_a^b(f) = \sup_P \sum_{i=1}^{n} \abs{f(x_i) - f(x_{i-1})},$ where the supremum runs over all partitions $P$ of $[a, b]$ given by $a = x_0 < x_1 < \cdots < x_n = b$ . The value $V_a^b(f)$ lies in $[0, \infty]$ .

Remarks

Example: for $f(x) = \sin x$ on $[0, 2\pi]$ , the total variation is $4$ . The function is monotone on $[0, \pi/2]$ , $[\pi/2, 3\pi/2]$ and $[3\pi/2, 2\pi]$ , and on each of these pieces the variation is the absolute change $1$ , $2$ and $1$ . Total variation is additive over adjacent intervals: $V_a^c(f) = V_a^b(f) + V_b^c(f)$ for $a < b < c$ .

Non-example of finite total variation: the indicator function of the rationals on $[0, 1]$ . A partition alternating between rational and irrational points with $n$ subintervals gives the sum $n$ , so the total variation is $\infty$ .

If $f$ is continuously differentiable on $[a, b]$ , then $V_a^b(f) = \int_a^b \abs{f'(x)}\, dx$ .

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Partition of an interval definition
Set bounded above definition
Supremum definition
Total variation definition

Tag	`ERHGXV009H`
URI	`m2://ERHGXV009H`
Kind	definition
Writing	W0 stub
MSC	26A45
Defines	Total variation
Uses	Partition of an interval, Supremum
Changed	2 November 2026

Total variation

Definition

Remarks

Reading path

Further reading

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