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