Show that a whole circle cannot be parametrized by a single map

Question

On page 4 of Guilleman and Pollack, the author asks, "Can you show that the whole circle cannot be parametrized by a single map?"

I get that the parametrization $\phi(\theta) = \left(\cos\theta,\sin\theta \right)$ fails because it is not continuous. Thus, it is not even a homeomorphism. But how would I prove the statement formally, rather than offer an example?

Thanks.

Answer 1

Hint: there can be no homeomorphism between an interval and a circle. Removing a point from the interior of an interval results in a disconnected space (with two connected components), but removing any point from a circle leaves a connected space (homeomorphic to an open interval via a restriction of the parametrization you mention in your question).