Let $f:ℝ→ℝ$ be a real analytic function. Assume that $f$ has infinitely may zeros. We define the equivalence relation $ℜ$ on $G(f)$ via $(x_1,f(x_1))ℜ(x_2,f(x_2))$ if and only if $f(x_1)=f(x_2)$. Here $G(f)$ is the graph of $f$.
My question is:
1) Find all the equivalence classes of $ℜ$.
2) Can we find an element $(a,b)∈G(f)$ such that the equivalence class of $(a,b)$ is finite? taking in account that $f$ is an analytic function.
The equivalence classes of $\Re$ correspond to the function values of $f$; there is one equivalence class for each function value of $f$, containing all points of the graph at which $f$ takes this function value.
Whether there are finite equivalence classes of $\Re$ is a roundabout way of asking whether there are values that $f$ takes finitely many times. There are analytic functions such as $f(x)=\sin x$ for which this is not the case, and analytic functions such as $f(x)=\mathrm e^{-x^2}\sin x$ for which this is the case.