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?



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