I am interested in the boundary behavior of Riemann maps. Is there a nice example of a region $G$ with Riemann map $\phi:\mathbb{D}\to G$ such that the limit of $\phi$ exists along some non-tangential paths but not along others?
Specifically, if the limit of $\phi(z)$ exists as $z\to 1$ along any (non-tangential) path above the real axis, but diverges along any path lying below the real axis.
If Riemann maps do not provide a nice example, how about any analytic function?