Does there exist such function?
I know that there is no homeomorphism between $(0,1)$ and $[0,1]$, because you can remove two points from $[0,1]$ and still have connected space and you can't do that in $(0,1)$.
Here however, we're looking at holomorphic functions and not homeomorphisms. Can I use the above result to conclude there is no such function?
A holomorphic function is open (the open mapping theorem) so $f[D(0,1)]$ is open (assuming $D(0,1)$ is the open disk) and so cannot be equal to the closed set $\overline{D(0,1)}$, by connectedness of $\Bbb C$, which tells us that the only subsets that are both open and closed are $\emptyset$ and $\Bbb C$.