Is the differentiability defined at the boundary points of a closed interval? If it is (e.g. the one-sided derivative), we can surely talk about the continuous differentiability. But somehow, in higher dimension, in most books, the differentiability is only defined at interior points of the domain. So for consistency, I prefer not to define the (continuous) differentiability at the boundary points.
May I change a bit of the Newton-Leibniz theorem like the this?
If given a function $F:[a,b]\to\mathbb{R}$, $f$ is continuous on $[a,b]$ and continuously differentiable on $(a,b)$, then $$F(b)-F(a)=\int_a^bF'(x)\text{d}x$$ This is somehow different because the the original requires the continuity of $F'(x)$ at $a,b$. But the thing is if the differentiability is not defined at the boundaries, we can not really say it is continuous there. But only the continuity on the open interval the function may not be integrable. E.g. For $F'(x)=\frac{1}{x}$ on (0,1), the Newton-Leibniz formula can not apply. But on the other hand, $F'(x)=\frac{1}{\sqrt{x}}$ on $(0,1)$, so we want the theorem apply here.
