Bounded Variation

A Function $f(x)$ is said to have bounded variation if, over the Closed Interval $x\in [a,b]$, there exists an $M$ such that

$$\vert f(x_i)-f(a)\vert+\vert f(x_2)-f(x_1)\vert+\ldots +\vert f(b)-f(x_{n-1})\vert \leq M$$

for all $a < x_1 < x_2 < \ldots < x_{n-1} < b$.