How to show something is not a regular surface?

Question

What is the general approach for showing that given equation does not represent a regular surface. For eg. if I have to prove that $y(x-a)+zx=0$ is not a surface, how do I approach?

Answer 1

The implicit function theorem tells you that if you consider the level surface $f(x,y,z)=0$, then the surface is a regular surface in a neighborhood of any point $(x_0,y_0,z_0)$ with $\nabla f(x_0,y_0,z_0)\ne 0$ [or use the derivative $df(x_0,y_0),z_0)$].

At points where this criterion fails (if any!), you need to ask yourself: Can I write this surface locally as a graph $z=\phi(x,y)$, $y=\phi(x,z)$, or $x=\phi(y,z)$ for a smooth function $\phi$? I've mentioned this criterion several times before; see this one, for example.