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    $\begingroup$ I doubt that there is or can be a criterion (for the existence of morphic continuation) of the generality you are looking for. Experience teaches us that one usually needs additional pieces of information like functional equation or integral representation 'ready' for regularization, or like in the case of the monodromy theorem, that there exists analytic continuation along the paths of curves in a simply connected domain containing $U$. $\endgroup$ Commented May 27, 2010 at 1:40
  • $\begingroup$ Hi Shenghao, for functions defined on open simple connected subset of $\mathbb C$, I heard once, that the analytic continuation can be made until the domain get surrounded by a dense set o singularities. (Perhaps someone else could give us a precise statement regarding to this fact) Even supposing this fact is true, for some class of holomorphic functions, It sounds for you as an acceptable criterion ? I am asking because frequently will be very hard to verify it. But if you have any interest on this kind of statement, I can look for you the details. $\endgroup$ Commented May 27, 2010 at 3:07
  • $\begingroup$ To Leandro: Thanks, and yes please, I'm interested in this. Thank you in advance. $\endgroup$ Commented May 27, 2010 at 4:38