How I can define an equivalence relation on the graph of a function?

Question

Let $a∈ℝ$ being fixed. Let $g:ℝ→ℝ$ be a real analytic function with infinitely many zeros. I want to classify the points $c,d$ such that $g(a)=(a-u)g′(c)≠0$ and $g(a)=(a-w)g′(d)≠0$ where $c∈(u,a)$ and $d∈(w,a)$ with $g(u)=g(w)=0$ and $u≠w,a≠u,a≠w$. I am thinking about an equivalence relation. However, I am not able to formulate it mathematically.