# Math Help - Borsuk-Ulam Theorem 1 dimensional

1. ## Borsuk-Ulam Theorem 1 dimensional

Let $f:S^1\to \mathbb{R}$ be a continuous map. Show there exists a point x of $S^1$ such that $f(x)=f(-x)$.

It says to define $g:S^1\to \mathbb{R}$ by $g(x)=f(x)-f(-x)$.

$g(x)=f(x)-f(-x)=-(f(-x)-f(x))=-g(x)$

By the Intermediate Value Theorem, there exist $x_0\in S^1$ such that $g(x_0)=0\Rightarrow f(x)=f(-x)$.

Here is my question: Why did we need to define a new function g?

2. ## Re: Borsuk-Ulam Theorem 1 dimensional

You defined the new function $g$ so that you could apply the Intermediate Value Theorem in a straightforward manner.

Did you have something else in mind?