f is bounded variation function iff there exist non-decreasing function g such that $f(x_2)-f(x_1) < g(x_2)-g(x_1)$ for $x_1 < x_2$
f is bounded variation function iff there exist non-decreasing function g such that $f(x_2)-f(x_1) < g(x_2)-g(x_1)$ for $x_1 < x_2$
Define $g(x) = \sup \left\{ {\sum\limits_{j = 1}^n {\left| {f(x_j ) - f(x_{j - 1} )} \right|} :\left\{ {x_0 ,x_1 , \cdots x_n } \right\}} \text{ is a partition of }[a,x]\right\}$