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In Barrett O'Neill's Elementary Differential Geometry book...he says that the cone

M:[itex]x^{2}[/itex] + [itex]y^{2}[/itex]=[itex]z^{2}[/itex] is not a surface in that

there exists a pointpin M such that there exists no proper patch in M which can cover a neighbourhood ofpin M

Intuitively I realise that this is point is the apex...here [itex]\left(0,0,0\right)[/itex]

but how would the patches for the rest of points be??...and what goes wrong with the patch at the apex of the cone??

Note: A PATCH IS A 1-1 REGULAR FUNCTION X:D→[itex]ℝ^{3}[/itex]..D open in [itex]ℝ^{2}[/itex]

This is an exercise problem.

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# Cone as surface

Can you offer guidance or do you also need help?

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