Throughout this question $I$ denotes the closed interval $[-1,1]$.

(a) For $n \in \mathbb{N}$, consider the $2 n+1$ points $r / n \in I$ with $r \in \mathbb{Z}$ and $-n \leqslant r \leqslant n$. Show that, if we colour them red or green in such a way that $-1$ and 1 are coloured differently, there must be two neighbouring points of different colours.

(b) Deduce from part (a) that, if $I=A \cup B$ with $A$ and $B$ closed, $-1 \in A$ and $1 \in B$, then $A \cap B \neq \emptyset$.

(c) Deduce from part (b) that there does not exist a continuous function $f: I \rightarrow \mathbb{R}$ with $f(t) \in\{-1,1\}$ for all $t \in I$ and $f(-1)=-1, f(1)=1$.

(d) Deduce from part (c) that if $f: I \rightarrow I$ is continuous then there exists an $x \in I$ with $f(x)=x$.

(e) Deduce the conclusion of part (c) from the conclusion of part (d).

(f) Deduce the conclusion of part (b) from the conclusion of part (c).

(g) Suppose that we replace $I$ wherever it occurs by the unit circle

$$C=\{z \in \mathbb{C}|| z \mid=1\}$$

Which of the conclusions of parts (b), (c) and (d) remain true? Give reasons.