Given that the Fourier coefficients of the function prove that the function is continuous

Question

Here is the problem:

Given that the Fourier coefficients of the function $f(x)$ are: $a_n=0$ , $b_n= \frac{1}{n!}$

1. Prove or disprove that $f(x)$ is continuous
2. Find the function explicitly
3. What are the Fourier coefficients of $f'(x)$