What is the possible behaviour on $\partial D$ of a continuous surjective map $D\to D$ continuously extended to $\bar D$?

Question

Let $D$ denote the open unit disk. If $f:D\rightarrow D$ is continuous, onto, and extends to a continuous function $\bar{f}:\bar{D}\rightarrow\bar{D}$, then must $\bar{f}(\partial D$) be contained in (or possibly even equal) $\partial D$?

Answer 1

The easiest to see is that $\bar{f}(\partial D)$ contains $\partial D$.

Indeed, let $y\in \partial D$ and let $y_n\in D$ converge to $y$.

• Since $f$ is onto there is a sequence $x_n$ such that $f(x_n)=y_n$.
• Since $\bar{D}$ is compact there is a subsequence $x_{\phi(n)}$ that is convergent, to some limit $x$.
• Since $\bar f$ is continuous, $\bar f(x)=y$
• Since $f(D)\subset D$, $x\not \in D$, therefore $x\in \partial D.$

Now can we imagine an example where $\bar{f}(\partial D)$ is not contained in $\partial D$? Yes:

• Consider the map $\bar f\colon \bar D\to \bar D$ defined by $z\mapsto z^2$, where $\bar D$ is regarded as the closed unit disk in $\Bbb C$. This is the closed unit disk, folded twice over itself with a branching point at $0$.

• Then use a homotopy to take a little bit of the boundary of one of the two sheets slightly back into the interior $D$.