\begin{definition}[Definition:Bounded Variation]
Let $a, b$ be real numbers with $a < b$.
Let $f : \closedint a b \to \R$ be a real function. 
For each finite subdivision $P$ of $\closedint a b$, 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:
:$\ds \map {V_f} P = \sum_{i \mathop = 1}^n \size {\map f {x_i} - \map f {x_{i - 1} } }$
We say $f$ is of '''bounded variation''' {{iff}} there exists a $M \in \R$ such that:
:$\map {V_f} P \le M$
for all finite subdivisions $P$.
\end{definition}