Consider the following set:
$$S := \{(x,y,z)\in \mathbb{R}^3 : x^2-4x+y^4+4y^3+6y^2+4y-z^4 = \alpha \}$$
The question asks to determine the value(s) of $\alpha \in \mathbb{R}$ such that $S$ is locally the graph of a $C^{\infty}$ function (of 2 variables). Doing this would require me to find the critical point of the stated expression, after finding this expression I would then plug this critical point back into my expression and I would obtain a value for $\alpha$. The reason this works is because the critical point is a singularity in my set, which is a surface.
Now the Implicit function theorem states that if I have a point $(a,b,c) \in S$ such that $F(a,b,c) = 0$ then I can express my surface near the point $(a,b,c)$ as the graph of a $C^{1}$ function i.e $z= f(x,y), y = f(x,z), x = f(y,z)$ (dependent on the specific results).
So in the set that I have above I only have the critical point that I solved for, but locally at this point I specifically cannot express my set as the graph of $C^{1}$ function.
I have two questions:
1) Is this the right way of thinking about this scenario?
2) If I did want to express my set as the graph of a function that would mean I have to somehow find some point $(a,b,c) \in S$ that would satisfy my expression. Visually I am fortunate enough to be able to see that the surface is smooth everywhere else except at the critical point as shown below. But say I didn't have the luxury of knowing what this visually looked like, are there any techniques to find said points or is it just a free for all?
