Is this map a homeomorphism onto its image?

The map $\phi$ is not a homeomorphism onto its image $S\subset{\mathbb R}^3$. The surface $S$ is an infinite vertical cylinder that intersects the $(x,y)$-plane in a curve which is part of a folium of Descartes, see the following figure:

This curve does not have a self-intersection, but almost. The inverse map $\phi^{-1}$ is well defined, but is not continuous at the points $\phi(0,y)=(0,0,y)$. Any neighborhood of $(0,0,0)$ contains points of $S$ that are mapped by $\phi^{-1}$ onto points in $(-1,\infty)\times{\mathbb R}$ of the form $(M,0)$ with $M\gg1$, hence lying far away from $\phi^{-1}(0,0,0)=(0,0)$.