Let $f:S^1\to\mathbb{R}$ be any continuous function where $S^1=\{(x,y)\in \mathbb{R}^2:x^2+y^2=1\}$ . What is the cardinality of this set $T=\{(a,b),a\in S^1,b\in S^1$ and $a\neq b:f(a)=f(b)\}$??
$S^1$ is connected and compact and continuous image of connected and compact is also connected and compact .I think it would be use here but don't know ...