On determining an infinitely divisible distribution function by its values on a half-line

Theorem. {\it Let $F(x)$ be an infinitely divisible distribution function with characteristic function $f(t)$. Suppose $f$ is holomorphic in $\{\operatorname{Im} z>0\}$ ($\{\operatorname{Im} z<0\}$). If an infinitely divisible distribution function $G$ coincides with $F$ on a half-line $(-\infty,a)$ (on a half-line $(a,\infty)$) then either $F(x)$ equals zero (equals one) on the half-line or $F(x)=G(x)$ for all $x$.}
The theorem generalizes a result of H. Rossberg [1]. Examples are given which show that the analiticity condition is essential.