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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?

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    the preimage is a regular surface iff the point is regular (by the implicit function theorem).2011-10-25

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