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let $f$ be any real valued continous function on $S^1$ ( unit circle in $\mathbb R^2$). does there exist uncountably many pairs of distinct elemens $x, y \in\mathbb R^2$ such that $f(x)=f(y)$?

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    $\begingroup$ anonymous: The title does not make sense. You are not asking about the existence of a subset of $\mathbb R^2$. $\endgroup$ Jan 18 '13 at 16:19
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HINT: If $f$ is constant, the result is trivial, so suppose that there are points $p,q\in S^1$ such that $f(p)\ne f(q)$. Now apply the intermediate value theorem to the two arcs of $S^1$ between $p$ and $q$.

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