Suppose that $f : \mathbb{R} \rightarrow \mathbb{R}$ is differentiable but $f \notin C^1 ( \mathbb{R} )$ . It means that $f'$ exist but it is not continuous.
Question 1 Is function $f'$ locally integrable. I.e. does there exist for every $a , b \in \mathbb{R}$
$$ \int_{a}^{b} f'(x) dx $$
I think, I should ask about existence of Lebesgue integral.
Question 2 If it exist, does the Newton-Leibniz formula holds?
$$ \int_{a}^{b} f'(x) dx = f(b) - f(a) $$
Comment. I am asking because I wanted to prove Cauchy's integral theorem using Stokes' theorem. One told me that I am not allowed to use Stokes' theorem if derivatives are not continuous.. So I wonder whether it is important. The simplest case of Stokes' theorem is Newton-Leibniz formula.
Q1: Since $f$ is differentiable on $(a,b)$ then $f(x):=\int_a^x{f'(t)dt}$, where $f'$ is absolutely continuous on $(a,b)$ (we can guarantee this case by Lebesgue theorem), so the answer is Yes $f$ is locally integrable in both senses Riemann and Lebesgue.
Q2: No cannot apply this. Since $f$ is not continuous on $(a,b)$. You may refer to the fundamental theorem of integral calculus.