I came up with a theorem.
Define a super function to be a function defined for all real numbers that is made out of sum/difference or product (not quotient) or composition of: sines,cosines,exponentials,polynomials.
For example, is a super function. Also is a super function. But is not because is not one of the basic functions on the list and also because division is not allowed in this defintion.
Theorem: Let be a finite interval and let be a super function. Then unless is identically zero the equation has only finitely many solutions on the interval .