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Independent increments and stationary increments make a Lévy process's characteristic functions multiplicative in time. Together with continuity of Lévy characteristic functions and the value one at time zero, this forces an exponential. A continuous local complex logarithm is additive: its failure of additivity would be a continuous integer multiple of $2\pi i$, hence zero. The Cauchy functional equation and subdivision then give the exponential at every time. This construction of the characteristic exponent of a Lévy process precedes the Lévy–Khintchine formula describing the exponent's possible form.

Fix $u\in \mathbb{R}$ and let $s\to t$. For any $\delta >0$, use $|e^{ia}-e^{ib}|\le \min (2,|a-b|)$ to obtainThe probability tends to zero by stochastic continuity. Taking the limit superior and then letting $\delta \downarrow 0$ provesThis 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.