From do Carmo:
Do Carmo says the forms $y =h(x,z)$ and $x=g(y,z)$ can be discarded since the projections of the cone over the $xz$ and $yz$ planes are not one to one. I do not understand why this is enough to discard these two functions.
2026-04-04 19:12:57.1775329977
Explanation for why the one sheeted cone is not a regular surface
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A graph $y=h(x,z)$ projects one-to-one onto (a portion of) the $xz$-plane, because for each point $(x,z)$ in the domain you have the single point $(x,h(x,z),z)$ on the surface.