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].

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