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If you have a differentiable complex function $f$ in a domain $D$, which is not identically zero, why are the zeros necessarily all of finite order and isolated?

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    If $f$ is differentiable, it is analytic (holomorphic). Assuming that the zeros of $f$ are not isolated, that means the set of zeros contains limit points. What can you now conclude about $f$ (this is a basic property of holomorphic functions)? Finally, if you have a zero which is not of finite order, then this zero would be a zero for every $n$th derivative of $f$. Can you conclude from this that $f$ must be identically zero? (think power series expansion).2012-03-31
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    @William Hmm, thanks. If the set of zeros contains limit points, does this imply that $f$ is identically zero on the some neighborhood of a some zero, but then by the Identity Theorem (as far as I know it's called) would then imply that $f$ is identically zero throughout the whole domain?2012-03-31

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