Definition:Total Variation

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

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

Let $X$ be the set of finite subdivisions of $\closedint a b$.

For each $P \in X$, write:


 * $P = \set {x_0, x_1, \ldots, x_{\size P - 1} }$

with:


 * $a = x_0 < x_1 < x_2 < \cdots < x_{\size P - 2} < x_{\size P - 1} = b$

Also write:


 * $\displaystyle \map {V_f} P = \sum_{i \mathop = 1}^{\size P - 1} \size {\map f {x_i} - \map f {x_{i - 1} } }$

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


 * $\displaystyle V_f = \sup_{P \in X} \paren {\map {V_f} P}$

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


 * $\map {V_f} P \le M$

for all $P \in X$.