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So I have been wondering the following

Is there a discrete subset $M$ of $\mathbb{C}$ and an entire function $f$ such that $f(M)$ is not a discrete subset of $f(\mathbb{C})$?

The definition I am using is that a set $M$ is called a discrete subset of $N \subset \mathbb{C}$ iff for all $z \in N$ there exists an $r > 0$ such that $M \cup K_r(z) \subset \{z\}$.

I have had some ideas trying it with $M = \mathbb{N}$ but that hasn't worked out yet. I'd appreciate any help!

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    $M = \mathbb{Z}$, $f = \sin$. More generally, for every entire transcendental function, we can find such $M$. – 2017-01-10
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    $M= \{ 2 \pi n + 1/n : n \in \Bbb{Z}^+ \}$ , $f= \sin$. – 2017-01-10

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