$f$ is a rational function $S^1\to S^1$, where $S^1$ is the one point compactification of $\mathbb R$. Is $\infty$ always a regular value of $f$?
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Once you've gone from $\Bbb R$ to $S^1$, does $\infty$ play any special role? – Hagen von Eitzen Dec 07 '15 at 07:27