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$f(x,y)$ is defined to be $\frac{x}{|y|}\sqrt{x^2+y^2}$ when $y \neq 0$ and $0$ when $y=0$. Is $f(x,y)$ continuous at $(x,y)=(0,0)$?

I don't know why, but I can't seem to find two paths that yield different limits at $(0,0)$.

The question asked to show that it is not continuous at the origin, but when I 3D graphed it, it appears to be continuous at the origin.

(This was a question asked on the real analysis final I just took. So it isn't homework, but seems to fall under that tag anyway since there is no tag for "finals")

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If $x=0$ and $y\rightarrow 0+$, the limit is 0. If $y=x^2$ and $x\rightarrow 0+$, the limit is $1$. If $y=x^4$ and $x\rightarrow 0+$, the limit is $+\infty$.

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  • $\begingroup$ Thank you! I must've botched the numbers the first time I tried that. $\endgroup$ – Taylor May 9 '13 at 0:36
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Hint: Pick some non-zero real constant $\alpha$ and set $$\alpha=\frac{x}{|y|}\sqrt{x^2+y^2}.$$ Solve this for $y$ under the assumption that $0<|x|<|\alpha|$ and $y>0$. That will give you a path along which you can approach the origin and get a different answer.

For a graph that should make it fairly clear that the function certainly shouldn't be continuous at the origin, check out the 3D plot here.

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