multivariable calculus - Describe the singularity of $\frac{xy}{x^2 + y^2}$ near $(x,y) = (0,0)$ - Mathematics Stack Exchange

in calculus class shown function $ f(x,y) = \frac{xy}{x^2 + y^2} $ not $c^\infty$ @ $(x,y) = (0,0)$. if exclude origin, can define 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 surface have @ origin? have been looking different names 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 origin consider $(x,y) = (r \cos \theta, r \sin \theta)$ , 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 long $r \neq 0$ there well-defined surface. there algebraic model kind of singularity?