Divergence of an integral over the set $(0,1) $ of a function with known behaviour at $1$

Question

Let $F$ be a continuous function on the open set $(0,1)$ and is such that $F(1)=\infty$ and $F'(1)=\infty.$ Is it true that $$ \int_a^1 F(z) dz =\infty $$ for every $a \in (0, 1)$? and if so, how can one show it?

Answer 1

No. There are functions that are integrable even if they and their derivatives diverge at a point. For example, $$f(x):=-\ln(1-x)\quad\textrm{(or }\ 1/\sqrt{1-x},\ldots)$$ $$\int_0^1-\ln(1-x)dx=\cdots=1$$