"Let $a ∈\mathbb C$ and $r > 0$.
Let $f : S(a, r) → R$ be continuous. Let $g : B(a, r) → R$ be the Poisson integral of $f$. Then
(1) $g$ is harmonic on $B(a, r)$ and
(2) $f \cup g : B(a, r) → R$ is continuous."
The second part regarding continuity is easy enough, but how should I handle the first part? It seems best to show that $g$ is the real part of a holomorphic function on $B(a,r)$ but how would I go about doing this? Any help is greatly appreciated.