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Does there exist a complex function $\,f(z)\,$ satisfying the following conditions:

(1) Continuous on the complex plane;

(2) $\,f'(0)\,$ exists;

(3) There are both an analytic point and a singularity in any punctured Neighberhood of $\,0\,$.

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Check the editing, please. – DonAntonio Sep 26 '12 at 12:01
up vote 3 down vote accepted

As it is phrased, $\frac{z^2-1}{z - 1}$ works or $\frac{\sin (z-1)}{z-1}$. It's continuous on $\mathbb{C}$ has a derivative at 0 and is analytic around z = 1/2 and has a removable singularity at z = 1.

A holomorphic function will not be continuous around poles or essential singularities.

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I think I made a mistake, In the last condition, the word "some" should be "any". In the "some" case, it is very easy to construct. Whatever, thank you again! – Riemann Sep 27 '12 at 2:39

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