The Dirichlet test for convergence of improper intervals strictly regards improper intervals on unbounded intervals. However, I can't seem to find any mention of a similar test for improper intervals of functions that are unbounded. If I follow the proof, I don't see any reason why it wouldn't work for unbounded functions.
The kind of statement I'm looking for is something along the lines of:
Let $f(x)$ be continuous and unbounded on $[a,b)$, and there exist some $M>0$ such that $|\int_a^c f(x)dx|\leq M$ for all $c\in (a,b)$. Additionally, let $g$ be positive, decreasing, and continuously differentiable on $(a,b)$, and $\lim\limits_{c\to b}g(x) = 0$. Then the improper integral $\int_a^b g(x)f(x)dx$ converges.
Is this correct? If not, where in the proof would it fail?