Let $[a,b]$ be a closed interval. A function $f\colon [a,b]\to \mathbb R$ is said to be of bounded variation if $$\sup_{n\in\mathbb N_{>0}}\sup_{a=x_0\lt x_1\lt\ldots\lt x_n=b}\sum_{j=1}^n|f(x_j)-f(x_{j-1})|<\infty.$$ We denote by $TV(f):=\sup_{n\in\mathbb N_{>0}}\sup_{a=x_0\lt x_1\lt\ldots\lt x_n=b}\sum_{j=1}^n|f(x_j)-f(x_{j-1})|$ the total variation of $f$, and we can endow the vector space of functions of bounded variation with the norm $\lVert f\rVert_{BV}:=TV(f)+|f(a)|$.

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