Is this function continuous at 0?

Asked Sep 24 '16 at 20:35

Active Sep 24 '16 at 20:35

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Let $f(x) = \frac{x^2y}{x+y}$ for $(x,y) \neq (0,0)$; $f(0)=0$

Is this function continuous at the origin? If I use polar coordinates, I find that it is continuous. But if I try the limit on the Ox axis and the line $y=x^3-x$, the limits yielded are different.

asked Sep 24 '16 at 20:35

Rainroad

1

This is not a continous function, it is not even defined for $x = -y$. How do you find it continous in polar coordinates? – Kaligule Sep 24 '16 at 20:54
lim as R goes to 0 of $\frac{R^2cos\theta sin\theta}{sin\theta+cos\theta} = 0$ – Rainroad Sep 24 '16 at 21:57
I see, but even then you cannot find an open neighbourhood of $0$ where $f$ is even defined. – Kaligule Sep 25 '16 at 08:45

Is this function continuous at 0?

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