Definition:Bounded Variation/Closed Bounded Interval
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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 $\closedint a b$.