in calculus class we are shown the function $ f(x,y) = \frac{xy}{x^2 + y^2} $ is not $C^\infty$ at $(x,y) = (0,0)$. However if we exclude the origin, we can define a surface:
$$ \left\{ \left( x,y,\frac{xy}{x^2 + y^2}\right) : (x,y) \in \mathbb{R}^2 \backslash \{ (0,0)\} \right\} \subseteq \mathbb{R}^2 \times \mathbb{R} $$
What kind of singularity does this surface have at the origin? I have been looking up different names but they are too technical:
- https://en.wikipedia.org/wiki/Du_Val_singularity
- https://en.wikipedia.org/wiki/Canonical_singularity
- https://en.wikipedia.org/wiki/Singularity_theory
Locally near the origin we could consider $(x,y) = (r \cos \theta, r \sin \theta)$ and our map looks like:
$$ (x,y) \mapsto \frac{{\color{#AAA}{r^2}} \sin \theta \cos \theta}{{\color{#AAA}{r^2}}} = \frac{1}{2}\sin 2\theta $$
so as long as $r \neq 0$ there is a well-defined surface. Is there an algebraic model for this kind of singularity?