I Know that $\Bbb S$\ $\lbrace-1\rbrace=\lbrace e^{i\theta}:\theta\in \left(-\pi,\pi\right)\rbrace=\lbrace e^{i\pi t}: t\in\left(-1,1\right)\rbrace$
Let $f:\Bbb S$ \ $\lbrace-1\rbrace\longrightarrow\left(-1,1\right)$ be a map where $e^{i\pi t}\mapsto t$
Why $f$ is continous function?
Can You help me please? or give me an Hint.