  Let $\Omega_1,\Omega_2$ be two disjoint open sets in $\mathbf C^n$ whose
boundaries share a smooth real hypersurface $M$ as relatively open subsets.
Assume that $\Omega_i$ is equipped with a complex structure $J^i$ which is
smooth up to $M$. Assume that the operator norm $\|J^2-J^1\|<2$ on $M$. Let $f$
be a continuous function on the union of $\Omega_1,\Omega_2, M$. If $f$ is
holomorphic with respect to both structures in the open sets, then $f$ must be
smooth on the union of $\Omega_1$ with $M$. Although the result as stated is
far more meaningful for integrable structures, our methods make it much more
natural to deal with the general almost complex structures without the
integrability condition. The result is therefore proved in the framework of
almost complex structures.
