I solved a problem in Calculus 2 exercise lesson and I do not fully understand it.
The task is to show that cross derivates differ in sign ( $f_{\text{xy}}(0;0)=-1$ and $f_{\text{yx}}(0;0)=1\, $ )
The function is: $ f(x,y)=\left\{ \begin{array}{cc} \frac{x y \left(x^2-y^2\right)}{x^2+y^2} & x^2+y^2\neq 0 \\ 0 & x^2+y^2=0 \end{array} \right. $
Short solution is:
I was told that lesson, what I should take away from this is, that cross derivates in this case
- are not continuous
- do not equal
Can someone explain:
- why it is not continuous
- why can't I just take $f_{xy}=\begin{cases} \frac{x^6+9 x^4 y^2-9 x^2 y^4-y^6}{\left(x^2+y^2\right)^3} & x^2+y^2\neq 0 \\ 0 & x^2+y^2 = 0 \end{cases} $ and where do I deduce, that I need to use limits at first place.