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Let $f(z)\in A(\mathbb D)$, where $A(\mathbb D)$ is the space of analytic functions on the open unit disk $\mathbb D$ and continuous on $\overline{\mathbb D}$.

Question: Is the connectivity of $f(\mathbb D)$ finite? If not, what condition should be added to make sure the connectivity of $f(\mathbb D)$ be finite?

Remark: when $f(z)$ be a conformal map or a proper map, the connectivity of $f(\mathbb D)$ is $1$.

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  • $\begingroup$ Could you please remind me how does one prove $f(\mathbb{D})$ is 1 when $f(z)$ is proper? $\endgroup$
    – Changyu Guo
    Commented Dec 21, 2013 at 22:54
  • $\begingroup$ Use the Riemann-Hurwitz Formula: $m-2=k(n-2)+r$ $\endgroup$
    – Jame Ake
    Commented Dec 22, 2013 at 2:25
  • $\begingroup$ However, we can prove that the “holes” of $f(\mathbb D)$ are Jordan domains with their boundaries. $\endgroup$
    – Jame Ake
    Commented Jan 3, 2014 at 3:58

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No, it can be infinite. Start with the function $w=f(z)$ mapping the unit disc onto itself. Let $I_k$ be disjoint closed arcs on the image disc in the $w$ plane. Deform the image region by adding two "horns" on each $I_k$ which go outside and then overelap. If the size of the horns tends to zero as $k\to\infty$, the function will be continuous.

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  • $\begingroup$ It's not clear on adding two "horns" on each $I_k$. Is $I_k$ closed curve or just closed arc? $\endgroup$
    – Jame Ake
    Commented Jan 4, 2014 at 8:09

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