Definition:Bounded Variation/Closed Unbounded Interval

Definition

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

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 $I$.