In his book page 250 Exer 11:
Suppose that $I=[a,b]$,$\Omega$ is a region ,$I\subset\Omega$,$f$ is continuous in $\Omega$,and $f\in H(\Omega-I)$,prove that actually $f\in H(\Omega)$.
If I follow the route Rudin proved SRP, I should prove that $Im(f)$ is a harmonic function in $\Omega$.But I don't know how to deal with it.
I will appreciate your help.