# Category:Bounded Variation

This category contains results about **functions of bounded variation**.

Definitions specific to this category can be found in Definitions/Bounded Variation.

## Definition

### Closed Bounded Interval

For each finite subdivision $P$ of $\closedint a b$, write:

- $P = \set {x_0, x_1, \ldots, x_n}$

with:

- $a = x_0 < x_1 < x_2 < \cdots < x_{n - 1} < x_n = b$

Also write:

- $\ds \map {V_f} {P ; \closedint a b} = \sum_{i \mathop = 1}^n \size {\map f {x_i} - \map f {x_{i - 1} } }$

We say $f$ is of **bounded variation** if and only if there exists an $M \ge 0$ such that:

- $\map {V_f} {P ; \closedint a b} \le M$

for all finite subdivisions $P$.

### Closed Unbounded Interval

For each finite non-empty subset $\mathcal S$ of $I$, write:

- $\mathcal S = \set {x_0, x_1, \ldots, x_n}$

with:

- $x_0 < x_1 < x_2 < \cdots < x_{n - 1} < x_n$

Also write:

- $\ds \map {V_f^\ast} {\mathcal S; I} = \sum_{i \mathop = 1}^n \size {\map f {x_i} - \map f {x_{i - 1} } }$

for $n \ge 1$, and $\map {V_f^\ast} {\mathcal S ; I} = 0$ otherwise.

We say $f$ is of **bounded variation** if and only if there exists an $M \ge 0$ such that:

- $\map {V_f^\ast} {\mathcal S; I} \le M$

for all finite non-empty subsets $\mathcal S$ of $I$.

## Also known as

Some sources say that $f$ is of **finite variation** rather than **bounded variation**.

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**bounded variation** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**bounded variation** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**bounded variation**

## Pages in category "Bounded Variation"

The following 12 pages are in this category, out of 12 total.