  Let $X, Y$ be two complex manifolds, let $D\subset X,$ $ G\subset Y$ be two
nonempty open sets, let $A$ (resp. $B$) be an open subset of $\partial D$
(resp. $\partial G$), and let $W$ be the 2-fold cross $((D\cup A)\times B)\cup
(A\times(B\cup G)).$
  Under a geometric condition on the boundary sets $A$ and $B,$ we show that
every function locally bounded, separately continuous on $W,$ continuous on
$A\times B,$ and separately holomorphic on $(A\times G) \cup (D\times B)$
"extends" to a function continuous on a "domain of holomorphy" $\hat{W}$ and
holomorphic on the interior of $\hat{W}.$
