Is the following set in $\mathbb R^2$ connected?

Is the following set in $\mathbb R^2$ connected?

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.