Let $X$ denote a real-valued random variable with distribution function $F$ and characteristic function $\phi$. Suppose that $\phi$ satisfies the following condition:
$$\lim_{T\to\infty}\int_{-T}^{T}\phi(t) ~dt = 2.$$
What can be said about the distribution?
Attempt:
The distribution is symmetric across the $x = 0$ axis (?)
The distribution is absolutely continuous.