(1)Let u be a bounded harmonic function on the upper half plane of $\mathbb{C}$. Show that $\forall y$ we have $u(x+iy)=\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{y\cdot u(t)}{(t-x)^2+y^2}dt$ for $x,y\in \mathbb{R}$.
(2)find a harmonic function on the upper half plane with lim$_{(y\to 0^+)}u(x+iy)=0,\ \text{if}\ x<0$ and lim$_{(y\to 0^+)}u(x+iy)=1,\ \text{if}\ x>0$
I have worked out the first part but I have no idea to construct such a function.
It is rather trivial if you have proven the first part. Just insert the function $$u(t)= \begin{cases}1 & t>0, \\ 0&t<0. \end{cases}$$ This works, as the integral representation only employs the function on the line $y=0$ where you know the limit. This is a quite typical example of a boundary value problem.
We thus have $$u(x+iy) =\frac{1}{\pi}\int_{0}^{\infty}\frac{y}{(t-x)^2+y^2}dt = \frac{1}{2} + \frac1\pi\arctan(x/y).$$