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Let us consider an equation $f(x)=0$ where $f$ is a real analytic function. Assume that this equation has still a finite number of solutions in any interval of the form $(y,2)$ where the number $y<2$ is located in an infinite discrete set. Can we deduce that the equation $f(x)=0$ has finite number of solutions in the interval $(-∞,2)$

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  • $\begingroup$ What we can deduce is that the number of the solutions of $f(x)=0$ is not uncountable $\endgroup$ – clark Aug 23 '13 at 20:30
  • $\begingroup$ @clark: I add the fcat that $f$ is analytic. $\endgroup$ – Germany Aug 24 '13 at 6:47
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No.

A very simple counterexample is $f(x)=\sin x$.

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