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Where can I find a proof/reference for the following fact?

Let $f$ be a holomorphic function with a zero of order $n$ at $z = 0$. Then for sufficiently small $\epsilon > 0$, there exists $\delta > 0$ such that for all $a$ with $0 < |a| < \delta$, $f(z) = a$ has exactly $n$ roots in the disc $|z| < \epsilon$.

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    I just found the question I was thinking of: http://math.stackexchange.com/questions/35304/proof-that-1-1-analytic-functions-have-nonzero-derivative. Duplicate?2012-01-23

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A reference is IV.7.4 in J.B. Conway's Functions of one complex variable.

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