BV (Bounded Variation) functions

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 $[a,b]$ .

Author: mathtuition88

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