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Suppose $f$ is analytic on $\mathbb{C}\setminus\{0\}$, with an essential singularity at $0$. Picard's Theorem asserts that $f$ will attain any value, with possible one exception, infinitely many times. Is it possible to have a function $f$ as above such that:

  1. at least one value is attained an uncountable number of times?
  2. all attainable values are attained an uncountable number of times?
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1 Answer 1

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No, not even 1.

Let $w\in\mathbb C$. Consider these sets: $A_n=\{z\in\mathbb{C}\,|\,n\leqslant|z|\leqslant n+1\}$ ($n\in\mathbb N$), as well as the sets $B_n=\left\{z\in\mathbb{C}\,|\,\frac1{n+1}\leqslant|z|\leqslant\frac1n\right\}$ (also with $n\in\mathbb N$). Each of them is compact and therefore, since $f$ is not constant, it follows from the identity theorem that the equation $f(z)=w$ has only finitely many solutions in each of them. Since the number of such sets is countable, the equation $f(z)=w$ has, at most, countable many solutions in $\mathbb{C}\setminus\{0\}$.

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