This is a theorem from my lecture notes:
If the continuous map $f: S^1 \to S^1$ extends to a continuous map $F: B(0,1) \to S^1$ the $f$ is homotopic to a constant map.
The proof just defines a homotopy $G$ by $G(z,u) = F(uz)$ so $G(z,1) = F(z) = f(z)$ for $Z\in S^1$ and $G(z,0) = F(0)$.
I don't understand why the criterion that $f$ ends to a continuous map $F: B(0,1) \to S^1$ is necessary? Surely you can still just define $G(z,u) = f(uz)$ to get the same result?