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==Formal statement==
:Let {{mvar|f}} be a [[holomorphic function]] on a [[Set (mathematics)|set]] {{math|''G'' \ ''K''}}, where {{mvar|G}} is an open subset of {{math|'''C'''<sup>''n''</sup>}} ({{math|''n'' ≥ 2}}) and {{mvar|K}} is a compact subset of {{mvar|G}}. If the [[Complement (set theory)|complement]] {{math|''G'' \ ''K''}} is connected, then {{mvar|f}} can be extended to a unique holomorphic function on {{mvar|G}}.
==Counterexamples in dimension one==
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