20
$\begingroup$

Prove that

  • if $f$ is of bounded variation in $[a,b]$, it is the difference of two positive, monotonic increasing functions; and
  • the difference of two bounded monotonic increasing functions is a function of bounded variation.

1 Answers 1

24
  • Let $f$ a function of bounded variation. Let $F(x):=\sup \sum_{j=1}^{n-1}|f(x_{j+1})-f(x_j)|=:\operatorname{Var}[a,x]$, where the supremum is taken over the $x_1,\ldots,x_n$ which satisfy $a=x_1
  • If $f$ and $g$ are of bounded variation so is $f-g$. If $f$ is increasing then we have, if $a=x_0