# Definition:Bounded Variation/Closed Bounded Interval

## Definition

Let $a, b$ be real numbers with $a < b$.

Let $f : \closedint a b \to \R$ be a real function.

### Definition 1

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 $P$.

### Definition 2

For each finite non-empty subset $\mathcal S$ of $\closedint a b$, write:

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

with:

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

Also write:

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

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

$\map {V_f^\ast} {\mathcal S; \closedint a b} \le M$

for all finite non-empty subsets $\mathcal S$ of $\closedint a b$.