Continuity of Green's function Suppose $\Omega \subset \mathbb C$ is a region (open and connected set) and let $$g(z,z_0)=G(z,z_0)-\log|z-z_0| $$ be its Green's function with pole at $z_0 \in \Omega$. Here $G(z,z_0)$ is the solution to the Dirichlet problem in $\Omega$ with boundary values $\log|\zeta-z_0|$.
I'm trying to solve the following problem (from Ahlfors):

Prove that $g(z,z_0)$ is simultaneously continuous in both variables, for $z \neq z_0$. Hint: Apply the maximum-minimum principle to $G(z,z_0)$.

I know that $\log|z-z_0|$ is simultaneously continuous in both variables for $z \neq z_0$, being the composition of continuous functions. Thus it remains to show the same for $G(z,z_0)$. So far I know that $G(z,z_0)$ is symmetric and harmonic in each variable.
How can I prove that $G(z,z_0)$ is simultaneously continuous in both variables for $z \neq z_0$? I can't see how to follow the hint unfortunately.
Thanks! 
 A: Let $\lvert z - z_0\rvert \geqslant 3\delta$, and suppose that $D_{3\delta}(z) \cup D_{3\delta}(z_0) \subset \Omega$. In the following, we shall assume that $\lvert w-z\rvert < \delta$ and $\lvert z_1 - z_0\rvert < \delta$. So we have $z_1,\,w \in \Omega$, sufficiently far from the boundary, and sufficiently far from each other. Then
$$\begin{align}
\left\lvert G(w,z_1) - G(z,z_0)\right\rvert &= \left\lvert \left(G(w,z_1) - G(w,z_0)\right) + \left(G(w,z_0) - G(z,z_0) \right)\right\rvert\\
&\leqslant \lvert G(w,z_1) - G(w,z_0)\rvert + \lvert G(w,z_0) - G(z,z_0)\rvert\\
& \leqslant \sup_{\zeta\in\partial\Omega} \left\lvert G(w,z_1) - G(w,z_0)\right\rvert + \sup_{\zeta\in\partial\Omega} \lvert G(w,z_0) - G(z,z_0)\rvert.
\end{align}$$
$G(\cdot\,,z_1) - G(\cdot\,,z_0)$ is the solution to the boundary values
$$\log \lvert\zeta-z_1\rvert - \log \lvert\zeta-z_0\rvert,$$
and $G(w,\,\cdot) - G(z,\,\cdot)$ is the solution to the boundary values
$$\log \lvert \zeta - w\rvert - \log \lvert \zeta - z\rvert.$$
Choosing $\lvert z_1-z_0\rvert$ and $\lvert w - z\rvert$ small enough, you can make the boundary values arbitrarily small.
