I am having trouble proving these two problems:
1) is $\{(x,y,z)\in \mathbb{R}|z=0, x^2+y^2\leq1\}$ a regular surface? I say no because the closed unit disk is a closed surface, so we cannot differentiate on the edges. But how can I define a function to prove this or is this enough explanation?
2) is $\{(x,y,z)\in \mathbb{R}|z=0, x^2+y^2<1\}$ a regular surface? I say yes because it is an open interval, so we can differentiate everywhere. But how can I define a function to prove this? Perhaps $f(x,y)=x^2+y^2$ as a level curve at $f=1$ and show that its gradient is nonzero? Also, saying $f=1$ wouldn't give me the correct closed interval, because it would contain points like (1,0) and (0,1), for example. Help!