Show that every homeomorphism $f: D^2\to D^2$ restricts to a homeomorphism $f_{|\partial D^2}: \partial D^2\to\partial D^2$
I want to proof the following statement. I have to show, that $f_{|\partial D^2}$ is a bijection and continuous and $f^{-1}_{|\partial D^2}$ is also continuous.
That $f_{|\partial D^2}$ is injective is clear, since $f$ is injective and we just observe $f$ on a subset of its preimage, but we would need that $f_{|\partial D^2}$ is surjective first.
How can I proof, that this function stays surjective? Let $y\in\partial D^2$. I have to find $x\in \partial D^2$ such that $f(x)=y$.
I tried to suppose, that $x\notin\partial D^2$ and then take on open set $U\ni y$ and tried to observe $f^{-1}(U)\ni x$ (open).
That $f_{\partial D^2}$ is continuous is implied by $f$ beeing continuous. So when I have that my function is bijective I can deduce it is a homeomorphism, since $\partial D^2$ is compact and Hausdorff.
What do you think?
Thanks in advance.