# BV functions of one variable

## Total variation

The total variation of a real-valued function $f$, defined on an interval $[a,b]$, is the quantity $\displaystyle V_a^b(f)=\sup_{P\in\mathcal{P}}\sum_{i=0}^{n_P-1}|f(x_{i+1})-f(x_i)|$ where the supremum is taken over the set $\mathcal{P}=\{P=\{x_0,\dots,x_{n_P}\}\mid P\ \text{is a partituion of }[a,b]\}$.

## BV function

$f\in BV([a,b])\iff V_a^b(f)<\infty$.

## Jordan decomposition of a function

A real function $f$ is of bounded variation in $[a,b]$ iff it can be written as $f=f_1-f_2$ of two non-decreasing functions on $[a,b]$.