Let $f:\mathbb{R}^{2}\rightarrow\mathbb{R}$ be a smooth ($C^{\infty}$) function and $O$ be an isolated critical point of $f$. I am looking at the local level sets diagram near $O$ from topological point of view. One easily finds 3 different cases:

1) local extremum $f=\pm(x^{2}+y^{2})$

2) saddle point $f=(y-x)(y-2x)...(y-nx)$, $n\geq2$ (for $n=2$ it is non-degenerate)

3) non-essential (remouvable) critical point $\ f=x^{2}+y^{3}$.

My question is whether there exist other possibilities different from the listed above.

I know that there is a quite tricky singularities theory, where critical points are classified from differential point of view, but here I am interested only in the local topological picture near $O$.


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