There exist a smooth real hypersurface $M\subset \mathbb C^2$, a point $p\in M$, and a function $u\in C^\infty(M;\mathbb C)$ such that $u$ is CR on $M$ and, for each of the two local sides of $M$ at $p$, there is no neighbourhood $U\subset \mathbb C^2$ of $p$ and no [holomorphic function](/page/Holomorphic%20Function) $F$ on the corresponding one-sided component of $U\setminus M$ whose boundary value on $U\cap M$ is $u|_{U\cap M}$.