Suppose $f:\mathbb R\times \mathbb R\to \mathbb R/\mathbb Z\times \mathbb R/\mathbb Z$ where the domain is given the usual topology and the latter the quotient topology. Why then is the restriction $f|_S:S\to f(S)$, where $S=\{(x,\sqrt{5} x): x\in \mathbb R\}$ not continuous with continuous inverse?
If I am not wrong, the quotient topology is comprised of sets whose preimages are open.
I can prove that $f|_S^{-1}$ is well-defined by showing that $f|_S$ is bijective, but I don't know why they are not continuous.