This is part of a series of questions of mine that came up when looking at the solutions to the resolvent equations related an eigenvalue problem I am working on. See for example that question.
My claim: if $f(x)$ is continuous everywhere such that $f(0)=0$, then $$ \int_0^1 \frac{|f(x)|}{x} dx <\infty. $$
I think it is true. I am just not sure how to prove it. But then, someone might prove me wrong by providing a counter-example.
This is not true. As a counter-example, let $f(x) = \lvert\log x\rvert^{-1}$ with $f(0) = 0$.