I am searching for sufficient conditions for the convergence of the Principal Value Integral $$I= \text{lim}_{\epsilon\rightarrow 0}\biggl[\int_{-\infty}^{-\epsilon} dx \frac{f(x)}{x}+\int_{\epsilon}^\infty dx \frac{f(x)}{x}\biggr] $$
I know that the Principal Value is a tempered distribution. So this limit exits when acting on Schwarz space.
I also know continuity at $x=0$ is too weak an assumption.
My question then is :
If f(x) is known to be :
a) Continuous everywhere on the real line
b) Absolutely integrable and
c) Uniformly bounded
Can I conclude this limit exists? If not, is it known what additional assumptions need to be made for this to exist? (Note in my case of interest, $f(x)$ is not necessarily differentiable.)
As a counterexample, define $f:\mathbb{R}\rightarrow\mathbb{R}$ piecewise by $f(x)=-\frac{1}{\log(x)}$ for $x\in(0,e^{-1})$, then $f(x)=\frac{1}{(ex)^2}$ for $x\in[e^{-1},+\infty)$ and $f(x)=0$ for $x\le0$. Then, $f\in L^1\cap L^\infty \cap C^0$ but from $$\lim_{\varepsilon\rightarrow0^+}\int_\varepsilon^{1/e}\frac{-1}{x\log(x)}=+\infty,$$ you get that the principal value integral in $0$ of $f$ is equal to $+\infty$.