Question
Let $\alpha:[a,b]\rightarrow\mathbb{R}$ be a function with finite total variation.
Let $F:\mathbb{R}\rightarrow\mathbb{R}$ be differentiable on [a,b].
Denote $\Delta\alpha(s):=\alpha(s)-\alpha(s-)$ where $\alpha(s-)$ is the left limit in s for $s>a$ and $\alpha(a-)=\alpha(a)$. Show that
$F(\alpha(x))=F(\alpha(a))+\int^{x}_{0}F(\alpha(s-))d\alpha(s)+\sum_{s\leq t}[F(\alpha(s))-F(\alpha(s-))-F'(\alpha(s-))\Delta\alpha(s )]$
Help! Any hints on where to start? And where/why does the sum come from?
