(i) Let $\gamma:[0,1] \rightarrow \mathbb{C} \backslash\{0\}$ be a continuous map with $\gamma(0)=\gamma(1)$. Define the winding number $w(\gamma ; 0)$ of $\gamma$ about the origin.

(ii) For $j=0,1$, let $\gamma_{j}:[0,1] \rightarrow \mathbb{C} \backslash\{0\}$ be continuous with $\gamma_{j}(0)=\gamma_{j}(1)$. Make the following statement precise, and prove it: if $\gamma_{0}$ can be continuously deformed into $\gamma_{1}$ through a family of continuous curves missing the origin, then $w\left(\gamma_{0} ; 0\right)=w\left(\gamma_{1} ; 0\right)$.

[You may use without proof the following fact: if $\gamma, \delta:[0,1] \rightarrow \mathbb{C} \backslash\{0\}$ are continuous with $\gamma(0)=\gamma(1), \delta(0)=\delta(1)$ and if $|\gamma(t)|<|\delta(t)|$ for each $t \in[0,1]$, then $w(\gamma+\delta ; 0)=w(\delta ; 0)$.]

(iii) Let $\gamma:[0,1] \rightarrow \mathbb{C} \backslash\{0\}$ be continuous with $\gamma(0)=\gamma(1)$. If $\gamma(t)$ is not equal to a negative real number for each $t \in[0,1]$, prove that $w(\gamma ; 0)=0$.

(iv) Let $D=\{z \in \mathbb{C}:|z| \leqslant 1\}$ and $C=\{z \in \mathbb{C}:|z|=1\}$. If $g: D \rightarrow C$ is continuous, prove that for each non-zero integer $n$, there is at least one point $z \in C$ such that $z^{n}+g(z)=0$.