Prove that $Ħ$ has no accumulation point

Prove that $Ħ$ has no accumulation point

Let $f,g,h,l:ℂ¨ℝ$ four continuous functions such that $f‚g$
and $h‚l$.
Let $D$ be an open set in $ℂ$. Let us define the set:
$$ƒ¶:{s=ƒ¿+iƒÀ¸D:f(s)=g(s),h(s)=l(s)}$$
My question is: Prove that $Ħ$ has no accumulation point, i.e., the set
$Ħ$ is discrete.