For questions about functions $f$ defined on an interval $[a,b]$ such that there exists a constant $M>0$, such that if $a=x_0<x_1<\ldots<x_n=b$, $n\in\mathbb N^*$, then we have $\sum_{k=1}^n|f(x_k)-f(x_{k-1})|\leq M$.
This concept can be generalized to infinite intervals, requiring that the constant is uniform.
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 $\lVert f\rVert_{BV}:=TV(f)+|f(a)|$.
