For $f(x,y) = \frac{xy}{x^2 + y^2}$ for $(x,y) \in \mathbb{R} - (0,0)$ and $f(0,0) = 0$, how do I prove that the function's partial dervatives are unbounded for $x^2 + y^2 < 1$? Since $x^2 + y^2 \geq 0$ for any real $x,y$, the case for (x,y) = (0,0) can be answered by the discontinuity of f at that point, and the problem is reduced to proved unboundedness for $0 < x^2 + y^2 < 1$.
I have tried making a sequence of the partial derivatives to prove that neither of them converge (thinking that convergence => boundedness would do the trick) by I wasn't able to involve the interval $0 < x^2 + y^2 < 1$, and I realized that non-convergence does not imply unboundedness.
The solution should not explicitly involve the calculation of the partials.