Question is to prove :

$f : S^1 \to \mathbb R$ is continuous then $f(x)=f(-x)$ for some $x\in S^1$

I guess it would be helpful to use intermediate value theorem

Assuming $f(x)\neq f(-x)$ then given any $p\in (f(-x),f(x))$ (assuming $f(-x)<f(x)$) there exists $y\in (-x, x)$ such that $f(y)=p$

I am not very sure of how to use this.. 

It would be helpful if someone can give some hint which would help me to solve this..

Thank you.