Let f : $S^1$ → R be continuous, where $S^1$ is the unit circle in $R^2$. (a) Show that there is a point z ∈ $S^1$ such that f(z) = f(−z). [z = (x; y), −z = (−x;−y)]\ (b) Show that f is not surjective.
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HINTS: $S^1$ is compact and connected. The map $S^1\to\Bbb R:z\mapsto f(z)-f(-z)$ is useful.
(a) Let $g(z)=f(z)-f(-z)$ and apply the Intermediate Value Theorem.
(b) You know that $S^1$ is compact. What do you know about the image of a compact space under a continuous map?