= Solution
The conclusions of (b) and (c) remain true. Indeed, the <unit circle> is a <path-connected space>, and the upper semicircle supplies a path from $-1$ to $1$; restricting the proposed closed cover or two-valued continuous function to that path reduces either claim to the corresponding result on $I$.
The fixed-point conclusion (d) fails. For example, any nontrivial <spatial rotation> $f(z)=e^{i\theta}z$ of the unit circle has no fixed point when $e^{i\theta}\ne1$; the antipodal map $f(z)=-z$ is the simplest example.
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