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Given ,

$$ f(x,y) = \begin{cases} \dfrac{xy^{3}}{x^{2}+y^{6}} & (x,y)\neq(0,0) \\ 0 & (x,y)=(0,0) \\ \end{cases} $$

We need to check whether the function is continuous at $(0,0)$ or not.. The solution says it is continuous at $(0,0)$.

What I tried was the following;

For the function to be continuous at the point $(0,0)$, the limit

$$\lim_{(x,y) \to (0,0)} f(x,y)$$

should exist.

Consider

$$\lim_{(x,y) \to (0,0)} \frac{xy^{3}}{x^{2}+y^{6}}$$

I choose a path $y=mx^{\frac{1}{3}}$ and approach $(0,0)$ along this path, thus the above expression becomes

$$\lim_{(x,y) \to (0,0)} \frac{xm^{3}x}{x^{2}+m^{6}x^{2}}$$

which comes out to be

$$\frac{m^{3}}{1+m^{6}}$$

Clearly the limit isn't unique and should not exist, but the solution says that the function is continuous at $(0,0)$.

How ? Can anyone help? What am I doing wrong ?

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You correctly proved that limit at origin doesn't exist. The answer key is wrong.

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