To discuss continuity of $\frac{xy^2}{x^2+y^4}$ at origin

Question

To discuss continuity of $$f(x,y) = \begin{cases}0 & \text{for } (x,y)=(0,0) \\ \frac{xy^2}{x^2+y^4}& \text{for }(x,y)\in\mathbb{R}^2\setminus\{0\}.\end{cases}$$

I converted to polars and my limit goes to 0. So i think it is continuous.I am not sure though

Thanks

Compute $f(y^2,y)$ for $y\not =0$. – Kelenner Jan 18 '16 at 09:45 — Kelenner, Jan 18 '16 at 09:45
https://math.stackexchange.com/q/93886/321264 – StubbornAtom Jan 05 '21 at 15:57 — StubbornAtom, Jan 05 '21 at 15:57

score 3 · Answer 1 · answered Jan 18 '16 at 09:47

Consider the parametrization $\Gamma_1(t): (0,\infty) \to \mathbb{R}^2$ by \begin{equation} \begin{split} x & = t^2\\ y & = t \end{split} \end{equation} the function is $$f(t^2,t) = \dfrac{t^2 t^2}{(t^2)^2 + t^4} = \frac{1}{2}$$ whose limit is $0.5$ when $t \to 0$. It is easy to find another parametrization $\Gamma_2(s)$ whose limit is $0$ when $s \to 0$. Therefore the function is not continuous.

To discuss continuity of $\frac{xy^2}{x^2+y^4}$ at origin

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