= Continuity of Lévy characteristic functions
{title2=$t\mapsto\varphi_{X_t}(u)$}
For a <Lévy process>, <stochastic continuity> gives continuity of its <characteristic function> in time at every fixed frequency. The elementary estimate $|\varphi_{X_s}(u)-\varphi_{X_t}(u)|\leq |u|\delta+2\mathbb P(|X_s-X_t|>\delta)$ proves this by first taking $s\to t$ and then $\delta\downarrow0$. The argument works for any stochastically continuous <stochastic process> and needs no finite-moment assumption.
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