Consider the multivariable function given by
$\begin{cases} \frac{xy(x^2 - y^2)}{x^2 + y^2} & (x, y) \neq (0, 0) \\ 0 & (x, y) = (0, 0) \end{cases}$
(a) Using the limit definition (and no other method), compute the partial derivatives $f_x(0, y)$ and $f_y(x, 0)$ for all $y$ and $x$ respectively.
Using any other method than the limit definition will result in a grade of zero.
(b) Use part (a) to demonstrate why $f_{xy}(a, b)$ and $f_{yx}(a, b)$ are not equal at every point $(a, b)$.
(c) If you are given the fact that $f(x, y)$, $f_x(x, y)$ and $f_y(x, y)$ are continuous for all $(x, y)$, what must be true as to not contradict the mixed partials theorem?
You do not need to prove your statement to part (c).
