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A thought just hit my mind that whatever branch points I have seen to make some functions holomorphic are always isolated i.e. there exists a neighbourhood around them which contains no branch points.

So I thought whether they are always isolated?

I looked up for a proof and even thought over it but couldn't find a suitable approach.

So my question is whether branch points are always isolated?

LoverOfAbstraction
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    Suppose you have branch points $a_1,a_2,a_3,\ldots$ and $\lim_n a_n = a$. What would it mean to say $a$ is a branch point? A curve winding once around $a$ would also wind around infinitely many of $a_1,a_2,a_3,\ldots$. There's no curve that winds only around $a$ and not around any branch points not equal to $a$. But maybe there's a standard definition that I don't know about. ${}\qquad{}$ – Michael Hardy Sep 10 '15 at 21:39
  • Yes. You are right that there won't be any curve winding around a alone but, if the sequence of branch points containing the complex numbers above converging to a lies on a segment, then we can draw a circle containing the segment totally and there might be other isolated branch points, so we cant be sure that the value of the function can't change right ? – LoverOfAbstraction Sep 10 '15 at 23:03

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