# Borsuk–Ulam theorem proof using Brouwer degree

I wonder if Borsuk–Ulam theorem (if $$f:\mathbb{S}^n\rightarrow\mathbb{R}^n$$ is continuous, then exists $$x_0\in\mathbb{S}^n$$ such that $$f(x_0)=f(-x_0)$$) can be sucesfully proved by using the Brouwer degree. My attempt is to find an homotopy from the function $$f(x)-f(-x)$$ to another suitable one in order to apply the invariance under homotopy of the degree and conclude that the degree of the considered function in a certain open set and in a certain point is not zero (which implies that the function $$f(x)-f(-x)$$ has a zero.

• You can get B-U this way for smooth maps $f$. This is in Guillemin and Pollack, maybe as an exercise. – Randall Oct 19 '18 at 3:14