Borsuk-Ulam theorem for $n = 1$ I'm thinking about the Borsuk-Ulam theorem for $n = 1$. How can I show that every continuous map $f : S^{1} \to \mathbb{R}$ has some $x \in S^{1}$ such that $f(x) = f(-x)$? 
My first idea was: I consider the new function $g = f(x) - f(-x)$. Let's now calculate the zero of $g$, i.e. $f(x) = f(-x)$. This equation is true when $f$ is the absolute value function. Is this the right approach?
Thanks in advance

Yes, it is. Now note that if $g(-x)=-g(x)$ and use the connectedness of $S^1$.
