Let $f : R → R$ be a differentiable function and $f(x) = 0$ for $|x| ≥ 10$. Let $g(x) = \sum_{k∈Z} f(x + k)$.

Question

Let $f : R → R$ be a differentiable function and $f(x) = 0$ for $|x| ≥ 10$.

Let $g(x) = \sum_{k∈Z} f(x + k)$.

Then one of the following is true:

(a) $g$ is differentiable and $g'$ has infinitely many zeros.

(b) $g$ is continuous and not differentiable,

(c) $g$ is differentiable and $g'$ has no zeros,

(d) $g$ is differentiable and $g'$ has only finitely many zeros.

I am able to show that $g$ is differentiable but cannot comment on the number of zeros of $g'$.