Here by a trigonometric series, I mean
$$ f(x)=\sum_{n=1}^\infty a_n e^{ i b_n x }, $$
where $a_n$, $b_n$ can be arbitrary complex numbers.
Two Questions:
Q1. Is it possible to make such a function $f$ convergent in $(0,1)$ but divergent in $(2,3)$?
Q2. As a related question, is it possible to make $f$ differentiable in $(0,1)$ but not differentiable in $(2,3)$?