Suppose f is a real-valued infinitely differentiable function and L_n is the set of limit points of the zero set of the nth derivative of f. Application of the Mean-Value Theorem yields that the sequence of sets {L_n} is nested upward. Let L be the union of all the sets L_n.

One can show that the set L_k consists of all real numbers for some nonnegative integer k if and only if the function f is a polynomial (of degree less than or equal to k).

My particular question is if we assume that the set L consists of all real numbers, must f also be a polynomial? Or can any other properties of f be determined?

I am interested in any thoughts, counterexamples, or further reading suggestions.