Let $f\colon[a,b]\to\mathbb{R}$ be differentiable on $(a,b)$. Suppose that the limits $f(a+)=\lim_{x\to a+}f(x)$ and $f(b-)=\lim_{x\to b-}f(x)$ exist and are finite.
My question is: Do we have $$\int_{a}^{b}f'(x)dx=f(b-)-f(a+)$$ without further assumption on $f$? If yes, what would be a reference for this result? If no, is there a counterexample for this?
Any help is highly appreciated.
No, this is not true. In fact, it's not even necessarily true that $f'$ is integrable. The classical example of such a pathological counterexample is Volterra's function.