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I am given a few functions and I have to study the following aspects:

  • Continuity in the point (0,0)
  • If the derivative exists at (0,0)
  • Continuity of the partial derivatives at (0,0)
  • Directional derivatives at (0,0)

One of the functions is, for example: $$ f(x,y) = \begin{cases} \frac{x^2y^2}{\sqrt(x^2+y^2)}, & \text{if if (x,y) not (0,0)} \\ 0, & \text{if (x,y) = (0,0)} \end{cases}$$

I was able to prove the continuity of the function via epsilon-delta proof (proved the $\lim\limits_{(x,y) \to (0,0)} f(x,y) = 0$), but my question is: do I always have to do this to prove the continuity of a function? So I have to do the same for the continuity of the partial derivatives?

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Use the polar coordinates transformation: $\begin{cases}x &=& r\cos\theta \\ y &=& r\sin\theta\end{cases}$

Then, $f(x,y) = r^3\cos^2\theta\sin^2\theta$

Now, since $\cos^2\theta\sin^2\theta$ is bounded, one can use the squeeze theorem.

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  • $\begingroup$ That is a cool way to prove it, but is it the only way (along with epsilon delta) to prove the continuity? $\endgroup$ – BSD Apr 17 '17 at 10:01

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