  If $R$ is a real analytic set in $\C^n$ (viewed as
  $\R^{2n}$), then for any point $p\in R$ there is a uniquely defined germ
$X_p$ of the smallest complex analytic variety which contains $R_p$, the germ
of $R$ at $p$. It is shown that if $R$ is irreducible of constant dimension,
then the function $p\to\dim X_p$ is constant on a dense open subset of $R$. As
an application it is proved that a continuous map from a real analytic CR
manifold $M$ into $\C^N$ which is CR on some open subset of $M$ and whose graph
is a real analytic set in $M\times \C^N$ is necessarily CR everywhere on $M$.
