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.

not sure...

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