Definition:Total Variation/Real Function/Closed Bounded Interval/Definition 1

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:


 * $\ds \map {V_f} {\closedint a b} = \sup_{P \in \map X {\closedint a b} } \paren {\map {V_f} {P ; \closedint a b} }$

This supremum is finite as, since $f$ is of bounded variation, there exists $M \in \R$ with:


 * $\map {V_f} {P ; \closedint a b} \le M$

for all $P \in X$.