Periodicity of an infinitely differentiable function

Question

Consider $f:[-\pi,\pi] \to \mathbb{C}$ be an infinitely differentiable function with $f^{(n)}(-\pi) = f^{(n)}(\pi)$ for all $n \in \mathbb{Z}^+$. Is this a periodic function ?

I think it is a periodic function but not sure.

I can only infer that its derivative is periodic: since for each $n \in \mathbb{Z}^+$, the derivative starts repeat itself at $\pi$, $f^{(n)}(-\pi) = f^{(n)}(\pi)$. but what happen for the original function (I mean without taking derivative, can I still say this function is a periodic function)?

Answer 1

Edit: This is the answer for the first version of the question.

If period means the smallest period the answer is: no. Constant functions are counterexamples.