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I am trying to rigorize the result that if f is 1-1 in a region R, then f'(z) is not zero in R.

This is what I have: Assume, by contradiction, that f'(zo)=0 for zo in R. Then

f can be expressed locally as :

f(z)=z^k.g(z)

for g(z) analytic and non-zero for some open ball B(zo,r)-{zo}

From this, we have to somehow use the fact that z^k is k-to-1, contradicting the

assumption that f is 1-1.

I don't know if we can use the fact that an open ball is simply-connected to

define a branch of log, from which we can define a branch of z^k, and then

conclude with the contradiction that f(z) is not 1-1.

Any suggestions for rigorizing?

Thanks.

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# If f(z) is 1-1, then f'(z) is not zero.

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