Give an example or say why it is impossible:
1. A function which is discontinuous at every point of the set but continuous elsewhere on
I think it's possible since that set could be written as a countable union of closed sets, but I can't think of a particular example.
2. An infinite subset of [0,1] with no limit points.
I think this is impossible but I'm not sure about my reasoning - is this correct?: Any subset of [0,1] is bounded so by the Bolzano Weierstrass theorem any sequence in this subset must have a convergent subsequence, so it has a limit point.
3. A closed set whose supremum is not a limit point of this set.
4.A power series that is absolutely convergent at only one point.
Probably something with the point being 0... not sure though
Thanks for any help