Definition:Total Variation/Real Function/Closed Bounded Interval/Definition 1
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Definition
Let $a, b$ be real numbers with $a < b$.
Let $f: \closedint a b \to \R$ be a function of bounded variation.
Let $\map X {\closedint a b}$ be the set of finite subdivisions of $\closedint a b$.
For each $P \in \map X {\closedint a b}$, 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 define the total variation $\map {V_f} {\closedint a b}$ of $f$ on $\closedint a b$ by:
- $\ds \map {V_f} {\closedint a b} = \map {\sup_{P \mathop \in \map X {\closedint a b} } } {\map {V_f} {P ; \closedint a b} }$
Sources
- 1973: Tom M. Apostol: Mathematical Analysis (2nd ed.) ... (previous) ... (next): $\S 6.4$: Total Variation: Definition $6.8$