Prove that $Ω$ has no accumulation point Let $f,g,h,l:ℂ→ℝ$ four harmonic functions such that $f≠g$ and $h≠l$. 
Let $D$ be an open set in $ℂ$. Let us define the set:
$$Ω:\{ s=α+iβ∈D:f(s)=g(s),h(s)=l(s)\}$$
My question is: Prove that $Ω$ has no accumulation point, i.e., the set $Ω$ is discrete. 
 A: Hints;
Suppose $\;w\in\Omega\;$ is an accumulation point of $\;\Omega\;$ , so that
$$\exists\;\{w_n\}_{n\in\Bbb N}\subset\Omega\;\;s.t.\;\;w_n\xrightarrow[n\to\infty]{}w$$
If we write $\;w_n=\alpha_n+i\beta_n\;,\;\;\alpha,\beta\in\Bbb R\;$ , we get
$$f(w_n)=g(w_n)\;,\;\;h(w_n)=l(w_n)\;,\;\;\forall\,n\in\Bbb N$$
Now read here , in particular the Improvement.
A: The assumptions do not imply that $\Omega$ is discrete in $D$. Let's assume for simplicity that $0 \in D$, and write $x = \Re z$ and $y = \Im z$. Then
$$\begin{align}
f(z) &= x,\\
g(z) &= xy,\\
h(z) &= xy^2 - \frac13 x^3,\\
l(z) &= xy^3 - x^3y,
\end{align}$$
are four different harmonic functions, but $D\cap i\mathbb{R} \subset \Omega$, that means $\Omega$ contains a nondegenerate line segment.
To obtain the conclusion that $\Omega$ has no accumulation point in $D$, you need stronger assumptions, for example that $h$ is a harmonic conjugate of $f$ and $l$ a harmonic conjugate of $g$, so that $\Omega$ is the zero set of the holomorphic function $m(z) = (f(z) + ih(z)) - (g(z) + il(z))$ which does not identically vanish by the assumption $f \neq g$.
