My question is the opposite to proving the existence of periodic solutions to ODE's.
Assume that $\ f(z)$ is a periodic function over the $\mathbb{R}$ , or doubly periodic over some lattice $\Lambda$ in $\mathbb{C}$. Where yes, $\ f(z)$ is smooth and continuous.
Can it be said that $\ f(z)$ is or must be a solution to some Ordinary Differential Equation? Or is periodicity not a strong enough attribute alone to assure the existence of an ODE?