# anyone help me with this problem, does there exist any subest of $\mathbb R^2$ such that

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 $f(x)=f(y)$?

• anonymous: The title does not make sense. You are not asking about the existence of a subset of $\mathbb R^2$. Jan 18 '13 at 16:19

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 $q$.