I was confronted with this question when reading "Stochastic Integration and Differential Equations" by Protter. Just after the definition of a Lévy process he says the following:
If $X_t$ is a Lévy-process and we consider the function $f_t(u)=\mathbb{E}(e^{iuX_t})$ where $f_0(u)=1$ and $f_{t+s}(u)=f_t(u)f_s(u)$, and $f_t(u) \neq 0$ for every $(t,u)$. Then, using the right continuity in probability we conclude that there exists a continuous function $\psi$ with $\psi(0)=0$ such that $f_t(u)=\text{exp}(-t\psi(u))$.
How can one prove this? (Right) continuity in probability seems a rather weak notion to me for the existence of a fully continuous $\psi(u)$. It would mean that also $f_t(u)$ is continuous right? So what we need is that the Fourier transform of a Lévy process is continuous i think. Any hints on that? (Probably its a well-known fact and I am missing something obvious here)
In the same section the so-called "Bochners Theorem" is also mentioned. Could anyone share a resource for me with the details and the sketch of proof?