Let $f:ℝ→ℝ$ be a real analytic function. We know that for any real number $a$, the fiber $f^{-1}(a)$ is a discrete set unless $f = a$. My question is: Find conditions on the function $f$ such that the fiber $f^{-1}(a)$ has a finite number of elements. Of course if $f$ is bijective then any fiber contain one element. I am interested in non bijective functions.
2026-03-28 02:06:39.1774663599
Find conditions on the function $f$ such that the fiber $f^{-1}(a)$ has a finite number of elements
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I'd wanted to post this as a comment, but it somehow became too long.
I can't give a full answer to your question, but at least I can give a partial answer. A sufficient condition to get a finite $f^{-1}(a)$ (for any $a$) is that the real analytic function is also a proper function. A function is called proper if the inverse of each compact set is compact again.
Singleton sets in $\mathbb{R}$ are compact sets, hence their inverse image is then both discrete and compact, hence finite. Unfortunately, I don't know about nice classifications of proper real analytic functions though, if there exists any at all.
Also note that this condition is sufficient for your problem but I'd be surprised if it were a necessary condition. Why is it, btw, that you're so interested in inverse images of points under real analytic functions? I've seen you ask several questions about them lately.