Can a non-constant continuous function be constant on these hyperbolas?

Can a non-constant continuous function $f:\mathbb{R}^2\to\mathbb{R}$ be constant on the following hyperbolas?$$H_a=\{(x,y)\in\mathbb{R}^2:x+1/y=a\},a\in\mathbb{R}$$$$H_\infty=\mathbb{R}\times\{0\}$$That is, such that $\forall a\in\mathbb{R}\cup\{\infty\}:P_1,P_2\in H_a\Rightarrow f(P_1)=f(P_2)$. I know that, if this were the case, we could write $f(x,y)=F(\frac{y}{1+xy})$ whenever $xy\neq-1$ with $F:\mathbb{R}\to\mathbb{R}$ continuous. But I can't arrive at any contradiction (if there's any).