$f(x,y)=\sqrt{xy}$
(i) Determine the maximal domain of $f$, thus that the range is real.
(ii) Is $f$ continuous its domain?
(iii) Determine and describe all Level sets of $f$
To (i) $D:=\{(x,y)\in \mathbb{R}^2:xy\geq0\}$
To (ii) $f$ is as a composition of continuous functions continuous everywhere on its domain.
To (iii) $\sqrt{xy} = c \Leftrightarrow xy=c^2 \Leftrightarrow y = \frac{c^2}{x}, x \neq 0$
Consider $x=0: \sqrt{0y}=c \Rightarrow c=0$
This seems very simple to me, but i can't see any mistakes. Is my solution correct?
i) You should specify that much more meaningfully. What parts of the plane is that?
ii) Be explicit: composition of what continuous functions?
iii) No, check your algebra and check your relation in different parts of the plane you identify in part i.