A New Elementary Proof of a Theorem of De La Valée Poussin

Abstract

We give a new elementary proof of the Classical Theorem: Let $f$ be of bounded variation on $[a,b]$ and let $V$ be its total variation function. Then there is a set $N$ such that $m\bigl (V(N)\bigr ) = m\bigl (f(N)\bigr ) = m(N) = 0$, and for each $x$ not in $N$, $f$ and $V$ have derivatives, finite or infinite, and $V^\prime(x) = |f^\prime(x)|$.