So I am working on the following problem. Let $U$ be a piecewise continuous function and bounded for all real numbers. Then define the Poisson Integral for the UHP to be (It can be deduce from the one for the unit circle).
$$P_U(z)=\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{y}{(x-\xi)^2+y^2}U(\xi)d\xi$$
Assume that $U$ has a jump at 0, for instance $U(+0)=0, U(-0)=1.$ Show that $P_U(z)-\frac{1}{\pi } \arg z$ tends to 0 as $z\rightarrow 0$.
The hint from my professor is consider $P_{U+V-V}=P_{U+V}-P_V$ where $V$ is some suitable function satisfies everything $U$ satiesfies. I guess we can show $P_U-\frac{1}{\pi}\arg z$ is in the form $P_{U+V}-P_V$ but how would we choose such $V$? How should I finish the problem with the aid of $V$?
Thanks