Is there a name or known properties of function sets closed under derivatives?

Question

I'm looking into the specific case of sets of Real functions closed under derivatives that don't contain zero such as {sin,cos,-sin,-cos}. Is there any information on such sets?

Answer 1

Take any continuously differentiable real function that is not a polynomial and take the set of all derivatives of this function.

This set will not contain zero since only constant functions have zero derivatives, and only polynomials eventually differentiate to constants. By definition it is closed under differentiation.

All the sets that you are describing will be unions of sets constructed in this way. I don't think these sets have a name.