(Solved): Consider the multivariable function given by \[ f(x, y)=\left\{\begin{array}{ ...

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Consider the multivariable function given by \[ f(x, y)=\left\{\begin{array}{ll} x y \frac{x^{2}-y^{2}}{x^{2}+y^{2}} & (x, y) \neq(0,0) \\ 0 & (x, y)=(0,0) \end{array}\right. \] (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 defintion will result in a grade of zero. (b) Use part (a) to demonstrate why \( f_{x y}(a, b) \) and \( f_{y x}(a, b) \) 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).