compactness and connectedness in R^2

i am just confused how to solve it.

Let A={(x, x Sin(1/x)) ∈ R²:x ∈ (0,1] } and
B=A U (0,0).
Then
1. is A compact and connected?
2. is B compact and connected?

Thanks in advance. Regards(Happy)(Happy)

Google:

PlanetMath: example of a connected space that is not path-connected