If $u$ is a harmonic function defined on an open set onto $\mathbb{R}$, and we know that there exists $x_0,y_0$ such that $u(x_0)+u(y_0)=M$ for some constant $M$, show that there exist infinitely many points $(x,y)$ such that $u(x)+u(y)=M$.
First I tried to use the mean-value property but as $M$ is not the maximum I cannot conclude anything by using inequalities as to prove the mean-value property itself. I also tried to prove that the set $A=\{(x,y) : u(x)+u(y)=M\}$ is open and closed and as it is nonempty (since $(x_0,y_0) \in A$) it should be the whole set where $u$ is defined (since it is a domain, so it is connected), BUT, I don`t know how to prove it is open, since I would need the mean-value property. If anybody could help me to solve this exercise...Thank you
Hint: For all small $r>0,$ there must exist a point $x_r\in \{x: |x-x_0| = r\}$ such that $u(x_r) = u(x_0).$ Otherwise the MVP does not hold at $x_0.$ Same thing at $y_0.$