[proofplan] We invoke the same-run Lewy-Henkin non-[extension theorem](/theorems/59), which supplies a smooth Levi-flat hypersurface in complex dimension at least two and a smooth CR function with no holomorphic extension to either local side near a chosen point. Taking the complex dimension to be $2$ gives the required hypersurface in $\mathbb C^2$. The remaining work is only to unpack the theorem's conclusion in the local-side language used in the statement. [/proofplan] [step:Apply the two-sided non-extension theorem in complex dimension two] By [citetheorem:9215] with $n=2$, there exist a smooth Levi-flat 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 $u$ does not extend holomorphically to either local side of $M$ near $p$. [guided] The input theorem [citetheorem:9215] is stated for smooth real hypersurfaces in $\mathbb C^n$ with $n\ge 2$. The present theorem asks for the special case $\mathbb C^2$, so we take $n=2$. The theorem then supplies exactly three objects: a smooth Levi-flat real hypersurface \begin{align*} M\subset\mathbb C^2, \end{align*} a point \begin{align*} p\in M, \end{align*} and a smooth complex-valued CR function \begin{align*} u\in C^\infty(M;\mathbb C). \end{align*} Its conclusion is that this function has no holomorphic extension to either of the two local sides of $M$ near $p$. Since a Levi-flat hypersurface is in particular a smooth real hypersurface, these objects satisfy the geometric hypotheses required here. [/guided] [/step] [step:Unpack the local-side conclusion] The phrase that $u$ does not extend holomorphically to either local side of $M$ near $p$ means the following. For each of the two connected local components of $U\setminus M$ determined by a sufficiently small neighbourhood $U\subset\mathbb C^2$ of $p$, there is no [holomorphic function](/page/Holomorphic%20Function) on that component whose boundary value on $U\cap M$ equals $u|_{U\cap M}$. This is exactly the non-extension conclusion supplied by [citetheorem:9215], so the objects $M$, $p$, and $u$ satisfy the formalized statement. [/step]