  Let $D\subset \C^n,$ $G\subset \C^m$ be pseudoconvex domains, let $A$ (resp.
$B$) be an open subset of the boundary $\partial D$ (resp.
  $\partial G$) and let $X$ be the 2-fold cross $((D\cup A)\times B)\cup
(A\times(B\cup G)).$
  Suppose in addition that the domain $D$ (resp. $G$) is {\it locally
$\mathcal{C}^2$ smooth on $A$} (resp. $B$). We shall determine the "envelope of
holomorphy" $\hat{X}$ of $X$ in the sense that any function continuous on $X$
and separately holomorphic on $(A\times G) \cup (D\times B)$ extends to a
function continuous on $\hat{X}$ and holomorphic on the interior of $\hat{X}.$
A generalization of this result for an $N$-fold cross is also given.
