This question already has an answer here:
Let $\bf S^1$ denote the unit circle in the plane $\bf R^2$. True/False ?
For every continuous function $f:\bf S^1 \to \bf R$, there exist uncountably many pairs of distinct points $x$ and $y$ in $\bf S^1$ such that $f(x)=f(y)$
Borsuk-Ulam or by taking the function $g(x)=f(x)-f(-x)$, IVT implies that there exist $x$ such that $f(x)=f(-x)$. But I'm unable to show the existence of uncountably many pairs. I think the fact $RP^1 \cong \bf S^1$ may be helpful. Any ideas?