Without using the Borsuk-Ulam theorem. Prove that any map $f\colon S^1 \to S^1$ mapping antipodal point to antipodal point has $\deg_2(f)=1$ by a direct computation.
I know that $f$ map anitpodal point to antipodal point, meaning $f$ is an odd function.
From one of my previous exercise, I have shown that for any smooth map $f\colon S^1 \to S^1$ there exist a smooth map $g\colon R\to R$ such that $f(\cos t, \sin t)=(\cos g(t), \sin g(t))$ and satisfying $g(t+2\pi)=g(t)+2\pi q$ for some integer $q$ and $\deg_2(f)= q \mod 2$
Since $f$ is an odd function, $f(-x)=-f(x)$ My professor says I need to use this fact to find a $g\colon R\to R$ such that $g(s+\pi)=g(s)+\pi q$ where $q$ is odd. But I have no clue how to do that.