So, I'd like to see conditions on a function $f(n,t)$ such that $F(t)$ from,
$$F(t)=\sum_n f(n,t)$$
Is continuous over a non-zero range, but is nowhere differentiable. The range of the summation could be anything, I don't really care.
My Conjecture
Such a function $f(n,t)$ only exists if it is periodic with respect to $t$.
I'd love to see a counter example :)