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

Moreover, for every $n\ge 3$, Henkin's degenerate mixed-Levi construction supplies a $C^\infty$ embedded real hypersurface $M_H\subset\mathbb C^n$, a point $p_H\in M_H$, and a function $u_H\in C^\infty(M_H;\mathbb C)$ such that $u_H$ is CR on $M_H$, the Levi form of $M_H$ at $p_H$ is degenerate, every neighbourhood of $p_H$ contains points where the Levi form has a positive eigenvalue and points where the Levi form has a negative eigenvalue, and $u_H$ has no local holomorphic extension to either side of $M_H$ at $p_H$. This second assertion is the existential conclusion of Henkin's special degenerate mixed-Levi models, not a consequence asserted for arbitrary hypersurfaces with non-degenerate mixed Levi signature.