Let $f:S^1 \rightarrow \mathbb{R}$ be a continous function. Show that exists $x\in S^1$ such that $f(x)=f(-x)$

Question

Since $f$ is continous, and $\mathbb{R} \setminus\{a\}$ is disconected for every $a \in \mathbb{R}$, but this doesn't hold for $S^1$, then exists $y \in \mathbb{R}$ such that $|f^{-1}(y)| > 1$. I'm not sure how to proceed to show that it must be at least an $x \in S^1$ such that $f(x)=f(-x)$