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Write down the biggest subset $D$ of $\mathbb C$ on which ${\rm Log(z)}$ is a continuous function. Explain why ${\rm Log(z)}$ is not continuous at points outside $D.$

Anyone know the answer to this?

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Related... – J. M. Apr 18 '12 at 14:35
My answer here:… might help... – Isaac Solomon Apr 18 '12 at 14:38

I'll tell you one possible answer, and you tell me why (I'm assuming the principal branch):

Answer: $D$ is the complex plane minus the non-positive reals.

Hint: Follow things in a semi-circle and observe the "jump" in angles.

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