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### Is composition of analytic function with invertible function analytic?

On $\mathbb{C}$ the map $z\mapsto \bar{z}$ is invertible. If $f\colon\Omega\to\mathbb{C}$ is analytic on $\Omega$ it is holomorphic, but the composition with $z\mapsto \bar{z}$ will not be holomorphic,...
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### Is composition of analytic function with invertible function analytic?

Take the following invertible function $f_w:\Bbb R\to\Bbb R$ with $w\in\Bbb Q\!\setminus\!\{0\}$: $$f_w(x) = \begin{cases} x, & x\in\Bbb R\setminus\Bbb Q \\ x+w, & x\in\Bbb Q \end{cases}$$ ...
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### Order of a zero and Identity Theorem

Take $g=0$. The negation of $iii)$ is: $\forall c ,\exists n$ such that $f^n(c)\ne0$. Since there exists such an $n$ for all $c$ and $n$ is a natural number, we can pass to the smallest such.
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