Show that f is not differentiable at the origin of the following function:
$f(x,y) = \left\{\begin{matrix}\frac{2xy}{x^2+y^2}, (x,y) \neq (0,0)\\ 0, (x,y) = (0,0) \end{matrix}\right.$
I was thinking that I would have to approach the origin from the left and right of the x and y-axis.
But given that it is a conditional function I have myself confused.
Could someone show me how to approach this question. Thanks !
If $y=mx$ we have $$f(x,mx)=\frac{2m x^2}{x^2+m^2 x^2}=\frac{2m}{1+m^2}$$ Then the limit depends by direction m and the function isn't continue. Therefore it isn't differentiable.