= Solution
Fix $u\in\mathbb R$ and let $s\to t$. For any $\delta>0$, use $|e^{ia}-e^{ib}|\leq\min(2,|a-b|)$ to obtain
$$
\begin{aligned}
|\varphi_{X_s}(u)-\varphi_{X_t}(u)|
&\leq\mathbb E|e^{iuX_s}-e^{iuX_t}|\\
&\leq |u|\delta+2\mathbb P(|X_s-X_t|>\delta).
\end{aligned}
$$
The probability tends to zero by <stochastic continuity>. Taking the limit superior and then letting $\delta\downarrow0$ proves
$$
\boxed{\varphi_{X_s}(u)\longrightarrow\varphi_{X_t}(u).}
$$
This proves <continuity of Lévy characteristic functions> from the elementary estimate, without requiring <moments> or replacing <convergence in probability> by an unjustified almost sure limit. At $t=0$, time approaches from the right. If stochastic continuity is formulated only at zero, <stationary increments> give the same argument at every $t$: the absolute value of $X_s-X_t$ has the law of $|X_{|s-t|}|$. For $u=0$ the <characteristic function> is identically one.
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