(a) Let $\gamma:[0,1] \rightarrow \mathbb{C} \backslash\{0\}$ be a continuous map such that $\gamma(0)=\gamma(1)$. Define the winding number $w(\gamma ; 0)$ of $\gamma$ about the origin. State precisely a theorem about homotopy invariance of the winding number.

(b) Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a continuous map such that $z^{-10} f(z)$ is bounded as $|z| \rightarrow \infty$. Prove that there exists a complex number $z_{0}$ such that

$$f\left(z_{0}\right)=z_{0}^{11}$$