Property of continuous functions defined on a sphere of R^n where n is odd

Question

If $B=B(0,1)$ is the unit open ball in $\mathbb{R}^n$ with $n$ odd and $f:\partial B \rightarrow \partial B$ is continuous, then exists $x\in \partial B $ such that $f(x)=x$ or $f(x)=-x$. Any idea of how to attack this problem using Brouwer degree?

Thanks in advance.