# Definition:Bounded Variation

## Definition

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:

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

We say $f$ is of bounded variation if there exists a $M \in \R$ such that:

$\map {V_f} P \le M$

for all finite subdivisions $P$.