Definition:Total Variation/Real Function/Closed Bounded Interval/Definition 2

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

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

Let $\map {\mathcal P_F} {\closedint a b}$ be the set of finite subsets of $\closedint a b$.

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 define the total variation $\map {V_f} {\closedint a b}$ of $f$ on $\closedint a b$ by:


 * $\ds \map {V_f} {\closedint a b} = \sup_{\mathcal S \in \map {\mathcal P_F} {\closedint a b} } \paren {\map {V_f^\ast} {\mathcal S ; \closedint a b} }$