Definition:Total Variation/Real Function/Closed Unbounded Interval

Definition
Let $I$ be an unbounded closed interval or $\R$.

Let $f : I \to \R$ be a real function. Let $\map {\mathcal P_F} I$ be the set of finite subsets of $\closedint a b$.

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:


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

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


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