This is my first mathematics related post. I'd be grateful for some suggestions on how to proceed.
Let be defined by . Show that this is a bijection but that its inverse is not continuous.
Attempt at solution:
Injective: Suppose . Then
This holds if and only if and .
From the first equality we have or .
What property of sin can I use in the second equality to show that the only simultaneous solution is x=y?
Surjective: Not quite sure where to start from here. I'm thinking if I let then would it be to say as the next logical step that and for some or do I have to justify this further?
Continuous: I know both and are continuous on . How can I use this to justify that is continuous?
Inverse not continuous: The hint I'm given is that this is not continuous because the pre-image of a small open set containing 0 is not open. How can there be a small open set containing 0 in the set ?
Any advice on how I can proceed would be much appreciated.