---
tag: F34S6P61SW
kind: definition
name: Bounded variation
msc: [26A45]
defines: A8251BHCMP
uses: [8QA5DEWNYD, 069P20KYQT]
sources:
  - url: https://en.wikipedia.org/wiki/Bounded_variation
---

## Definition

Let $a < b$ be real numbers and let $f \colon [a, b] \to \R$ be a function.
For a [[partition|069P20KYQT]] $P$ given by $a = x_0 < x_1 < \cdots < x_n = b$, put
$$V(f, P) = \sum_{i=1}^{n} \abs{f(x_i) - f(x_{i-1})}.$$
The [[total variation|8QA5DEWNYD]] of $f$ on $[a, b]$ is $V_a^b(f) = \sup_P V(f, P)$, where the supremum runs over all partitions of $[a, b]$.
The function $f$ has *bounded variation* on $[a, b]$ if $V_a^b(f) < \infty$.

## Remarks

Example: a nondecreasing function $f$ on $[a, b]$ has bounded variation.
For every partition the sum telescopes: $V(f, P) = \sum_i (f(x_i) - f(x_{i-1})) = f(b) - f(a)$, so $V_a^b(f) = f(b) - f(a)$.

Non-example: let $f(0) = 0$ and $f(x) = x \sin(1/x)$ for $0 < x \le 1$.
The function $f$ is continuous, but it does not have bounded variation.
At the points $x_k = 1/((k + \frac12)\pi)$ the values are $\pm x_k$ with alternating signs, so a partition through $x_1, \dots, x_N$ gives $V(f, P) \ge \sum_{k=1}^{N} x_k$, and this sum diverges like the harmonic series.

The set of functions of bounded variation on $[a, b]$ is often written $BV([a, b])$.
Every function of bounded variation is bounded, since $\abs{f(x)} \le \abs{f(a)} + V_a^b(f)$.