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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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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