Consider the function $h(x,y)=(x−y^2)(x−3y^2)$, $(x,y)\in \mathbb{R}^2$.
Show that the set $\{(x,y) | h(x,y) = 0\}$ cannot be expressed as the local graph of a $C^1-$function over the $x-$axis or the $y-$axis near the origin.
I have no idea where to begin. I can only draw a graph and say that it fails the vertical line test. How to write it in a rigorous manner?
I have been taught Implicit Function Theorem. So, probably a hint along those lines.
You can prove that $0$ is a regular value for a $C^\infty$ function if and only if the locus of that $C^1$ function is locally the graph of a real $C^1$ function on the $x-$ axis or the $y-axis$. So you must prove that $0$ is not a regular value for $h$ that means that
$\nabla(h)(x,y)\neq 0$ for all $(x,y)\in Z(h)$
but in $0$ your function is such that
$\nabla(h)(0)=0$