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Prove that $\dfrac{2(x^3+y^3)}{x^2+2y}$ is discontinuous at $(0,0)$.

Also show that the partial derivatives with respect to $x$ and $y$ at $(0,0)$ exist.

egreg
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user119065
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    The expression is undefined at $(0,0)$, so you need to add a definition for the function at that point. In the present version, asking whether the partial derivatives exist has no sense. – egreg Jan 04 '14 at 14:02

1 Answers1

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Hint

As mentioned by Greg the function is undefined at $(0,0)$ so you should add the value of $f$ at this point. Let's assume that $f(0,0)=0$. Prove by taking the limit in the direction $y=-\frac{x^2+x^3}{2}$ that $$\lim_{(x,y)\to(0,0)}f(x,y)$$ doesn't exist.

By the definition $$\frac{\partial f}{\partial x}(0,0)=\lim_{x\to0}\frac{f(x,0)-f(0,0)}{x}\quad;\quad\frac{\partial f}{\partial y}(0,0)=\lim_{y\to0}\frac{f(0,y)-f(0,0)}{y}$$