# Find sufficient and necessary conditions in which the equation has finite number of solutions

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 $(−∞,2)$.

• 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., $sin(1/x)$ – DBFdalwayse Aug 24 '13 at 7:00
• @DBF: I assume that $f$ is not a polynomial. – Germany Aug 24 '13 at 7:04
• then being monotonic is sufficient, but not necessary. – DBFdalwayse Aug 24 '13 at 7:09
• 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. – Gerry Myerson Aug 24 '13 at 7:13