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The motivation to this question can be found in: Can we deduce that the equation $f(x)=0$ has finite number of solutions in the interval $(-∞,2)$?

My question is: Assume that $f$ is not a polynomial, then: Find sufficient and necessary conditions in which the equation has finite number of solutions in the interval $(−∞,2)$.

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    $\begingroup$ If f is a polynomial of degree n, then f can have at most n zeros in the real line. If f is monotonic, non-zero, then it will have at most one solution. But you can have infinitely-many with, e.g., $sin(1/x)$ $\endgroup$ – DBFdalwayse Aug 24 '13 at 7:00
  • $\begingroup$ @DBF: I assume that $f$ is not a polynomial. $\endgroup$ – Germany Aug 24 '13 at 7:04
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    $\begingroup$ then being monotonic is sufficient, but not necessary. $\endgroup$ – DBFdalwayse Aug 24 '13 at 7:09
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    $\begingroup$ If its derivative (or 2nd derivative, or third derivative, ...) has only finitely many zeros in the interval, then $f$ has only finitely many. I think the only necessary and sufficient condition is that the equation should have only finitely many solutions in the interval. $\endgroup$ – Gerry Myerson Aug 24 '13 at 7:13

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