How do you prove that the real part of the characteristic function of the continuous probability distribution $f(x)$ is a characteristic function, but the imaginary part is not?
The second part is simple, and, if I am correct, the solution is below: ${\mathop{\rm Im}\nolimits} \left( {{f_\xi }(0)} \right) = 0$, and it is impossible for the correspondent candidate for characteristic function $h(t) = {\mathop{\rm Im}\nolimits} \left( {f(t)} \right)$.
But could you help, please, how to prove the first part? There is a solution through Bochner’s theorem, but it seems there is a better solution.
Thank you in advance!