I mean, the easiest way would be to note that $\displaystyle f:\mathbb{R}^2\to\mathbb{R}$, $\displaystyle f(x,y)=x^2+y^2$ is continuous (it's just a polynomial) and $\displaystyle A=f^{-1}({1})$ and for the second one are you sure it isn't closed?
oho....actually the 2nd one i think its a hyperbola...so it should be not closed...am i wrong?.....but in the first case...i did not understand what r u actually doing to prove that set A is closed