This is (a part) of exercise $7c$ page $66$ of DoCarmo's book (Differential geometry of curves and surfaces).
Let $f(x,y,z)=xyz^{2}$. I'm trying to figure out if the preimage of $f$ under $0$ is a regular surface.
Basically the preimage is the union of the three coordinate planes in the $xyz$ plane but how can we prove that there exists a point in where regularity fails?
I don't see this very clear. Can you please help?