TITLE: Is the following set in $\mathbb R^2$ connected?
QUESTION [5 upvotes]: Is the following set in $\mathbb R^2$ connected?: $A=\{(x, y) : x^2y^2 = 1\}$

My attempt: $f:\mathbb R^2\to\mathbb R:(x,y)\mapsto xy$ is continuous and $f(A)=\{-1,1\}.$ Since $f(A)$ is disconnected so is $A.$
Please tell me if it's correct or not.

REPLY [5 votes]: To cut down on un-answered questions, I'll simply make Daniel Fischer's comment an answer.
"If we're picky, you need to prove that $f(A)=\{-1,1\},$ say by observing $\{(1,-1),(1,1)\}\subset A.$ But aside from that, your proof is correct and elegant."