Let us define a function $f:\mathbb{R}→S^1$ by $f(x)=exp(ix)$, where $S^1=\{z∈\mathbb{C}:|z|=1\}$. Prove that $f$ is a surjective continuous function. Also show that $f^{-1}$ is continuous in an open neighborhood of $S^1$.
It is clear that the function is well defined and surjective also, as for any element $z$ in $S^1$ is of the form $exp(ix)$, for $x$ in $\mathbb{R}$. So it is surjective function. But I am stuck with the continuous and inverse continuity of $f$ in an open neighborhood. For continuity, for a chosen $e>0$ there exists a $r>0$ such that if for $x_0∈\mathbb{R}$ with $|x-x_0|<r$ then we have to show $|f(x)-f(x_0)|<e$. That is $|exp(ix)-exp(ix_0)|<e$. How can I show this. Also how can I find an open set where $f^{-1}$ is continuous. Please help me to solve this.