So, I saw the following question concerning complex analysis in one variable:
Can you find a continuos, surjective function $f:D\rightarrow D$, where $D$ is the closed unit disk $\overline{B_1(0)}\subseteq \mathbb{C}$, such that the following holds?
$f$ is holomorphic on the open unit disk $B_1(0)$, but there is no point $z$ on the boundary $\partial B_1(0)$ with the property that there is an open neighborhood $U$ and a holomorphic function $g:U\rightarrow\mathbb{C}$, such that $g$ and $f$ agree on $U\cap B_1(0)$.
I have already found a power series which has the latter property, but is sadly only defined on the open unit disk. However, I have no clue whether such a function can be found at all.
How can I tackle that exercise? Might Brouwer's fixed-point theorem be somehow useful here?