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Does the identity principle hold for meromorphic functions? (The identity principle states that if $Q$ is a connected set and $f(z)= g(z)$ for all $z$ in some subset $A$ of $Q$ which has limit points in $Q$, then $f(z) = g(z)$ for all $z$ in $Q$)

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Use the identity principle on the complement of the poles of both functions. –  Qiaochu Yuan Sep 2 '12 at 22:19
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So then set is still connected, but what if in removing the poles, you remove the limit point of the subset, or if at a point one function has a removable singularity and the other has a pole? –  confused Sep 2 '12 at 22:30
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Work with $f - g$. If $f$ and $g$ agree on a set with a limit point, then the limit point is a removable singularity of $f - g$. ("Poles of both functions" should have been "poles of either function" above; I meant for "both" to be applied to "poles" and not to "functions.") Also, you want $Q$ to be open. –  Qiaochu Yuan Sep 2 '12 at 22:41
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