Take a continuous, antipodal function $f: S^1\to S^1$ where antipodal means $$f(-x)=-f(x)$$ for all $x\in S^1$. How to prove that the winding number of $f$ is not equal to zero? This is my intuition but I failed to find a proof.
By $S^1$ I mean a circle in the plane $\bf C$ around zero. By winding number I mean the winding number of the graph of $f$ in the plane around zero. It is an integer representing the total number of times that the graph of $f$ travels counterclockwise around zero.
I would be interested in both an analytical proof as well as an algebraic topology proof (if this exists).
Thank you in advance for your help!
EDIT: I want to thank the people who gave comments and asked questions to improve the quality of the question. I tried to reflect them in this updated version of the question. Thank you also for the fixed point answer.