# continuity of isomorphism of unit circle

I try to show $$\mathbb{S}^1\cong[0,1)$$, by the map $$f(x) = (\cos2\pi x,\sin2\pi x)$$, for $$x\in[0,1)$$. It's clear that $$f$$ is continuous and bijective. But I don't know how to show the inverse map $$f^{-1}$$ is continuous also. Any ideas?

• It isn't.${{}}$ – Lord Shark the Unknown Apr 11 at 19:34
• So what the isomorphic mean here? Just need $f$ to be continuous and bijective? – QD666 Apr 11 at 19:36
• Show that $f^{-1}$ is discontinuous by considering what $f^{-1}$ does near the point $(1,0)$. – kccu Apr 11 at 19:39
• We can't say what "isomorphic" means here - you introduced the word in your question. Where did you get it from? – Rob Arthan Apr 11 at 20:06
• it's from an introduction to manifold class. The professor never actually mentioned about what "isomorphic" he refers to? I think I should probably ask him. Thanks anyways – QD666 Apr 11 at 20:10