A problem on continuous functions
Solution 1:
If $f$ is a constant map, then there is nothing to prove. Otherwise, there exist $p,q\in S^1$, such that $f(p)<f(q)$. Denote the two open arcs joining $p$ and $q$ by $I$ and $J$. Since $I$ and $J$ are connected and $f$ is continuous, $f(I),f(J)\supset(f(p),f(q))$. It follows that for every $t\in(f(p),f(q))$, there exist $x\in I$ and $y\in J$, such that $f(x)=f(y)=t$, which completes the proof.