Let $F: D^2 \to D^2$ be a continuous application such that $F (S^1) \subset S^1$. Define $$ f: S^1 \to S^1: z \mapsto F(z). $$
Prove that $F$ is surjective or $f$ is homotopic at a constant.
Recall that $D^2 = \{x \in \Bbb R^{n + 1}: \| x \| \le 1\}$.
I can not express restriction of $F$ along with the continuous extension of $f$, I also have problem with contradomines because they are different.
Thanks