TITLE: Continuous function on unit circle has fixed point
QUESTION [6 upvotes]: The question I have is:
Let $f: S^1 \rightarrow S^1$ be a continuous function, where $S^1$ is the unit circle. Prove that if $f$ is not onto, then $f$ must have a fixed point.

REPLY [4 votes]: Let $x\in S^1$ be not in the range of $f:S^1\to S^1$, and let us introduce a (constant-speed) parameterization $\phi:[0,1)\to S^1$ such that $\phi(0)=x$. Then we consider $F=\phi^{-1}\circ f\circ \phi:[0,1)\to[0,1)$. We can continuously extend $F$ to $F:[0,1]\to[0,1]$, and obviously, we have $F(0)>0$ and $F(1)<1$. Hence $F$ has a fixed point.