Let

$$B_{r}(0)=\left\{(x, y) \in \mathbb{R}^{2}: x^{2}+y^{2}<r^{2}\right\},$$

$B=B_{1}(0)$, and

$$C=\left\{(x, y) \in \mathbb{R}^{2}: x^{2}+y^{2}=1\right\}$$

Let $D=B \cup C$.

(i) State the Brouwer fixed point theorem on the plane.

(ii) Show that the Brouwer fixed point theorem on the plane is equivalent to the nonexistence of a continuous map $F: D \rightarrow C$ such that $F(p)=p$ for each $p \in C$.

(iii) Let $G: D \rightarrow \mathbb{R}^{2}$ be continuous, $0<\epsilon<1$ and suppose that

$$|p-G(p)|<\epsilon$$

for each $p \in C$. Using the Brouwer fixed point theorem or otherwise, prove that

$$B_{1-\epsilon}(0) \subseteq G(B)$$

[Hint: argue by contradiction.]

(iv) Let $q \in B$. Does there exist a continuous map $H: D \rightarrow \mathbb{R}^{2} \backslash\{q\}$ such that $H(p)=p$ for each $p \in C$ ? Justify your answer.