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

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

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

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

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