Real analysis/Total Variation

Let f(x) be a continuous function on an interval [a,b]. A partition of f(x) on the interval [a,b] is a sequence x_k such that x₀=a, x_k>x_k-1, and such that x_n=b. The total variation t of a function on the interval [a,b] is the supremum

t= sup{a|a=$\sum _{k=1}^n\Vert f(x_k)-f(x_{k-1})\Vert$ and x_{k<?sub> is a partition of [a,b]}.}

If this supremum exists, the the function is of bounded variation on [a,b]. If a real function is of bounded variation over its whole domain, then it is called a function of bounded variation.