In this paper, we show that if  f  is a separately holomorphic function on X \ P, where $X := ExV\capU xF$ with $U \subset C^m$ and $V \subset C^m$ are domains, $E \subset U$ and $F \subset V$ are locally pluriregular set, and P is a closed pluripolar set in an open neighborhood W of X, then there is an open neighborhood $\Omega$ of X, a closed pluripolar set S in $\Omega$ and a function $\hat{f} \in \theta(\Omega \ S) such that $S\cup X* \subset P$ and$ \hat{f}|_{x*\P} = f|_{x·\P}$ for some subset X* of X so that X \ X* is pluripolar.