How to prove that the set is not a regular surface?

I know that $$S=\{(x,y,z)\in \mathbb R^3: z^2=x^2+y^2\}$$ is not a regular surface, bacause it has a vertex in $(0,0,0)$. But how to show it precisely? Maybe here is usefull the theorem that a regularv surface is locally a graphic of infinite differentiable function of the form $z=f(x,y)$ or $y=g(x,z)$ or $x=h(y,z)$?

Thanks

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