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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 !


marked as duplicate by Arnaud D., Cesareo, Claude Leibovici, user416281, Deepesh Meena Sep 20 '18 at 18:42

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ Try approaching along other lines: $x=y$ is a good start. $\endgroup$ – James Mar 24 '15 at 19:13
  • $\begingroup$ @ArnaudD. It is a duplicate, but if it matters, this one is 5 months older than the duplicate you've pointed to. $\endgroup$ – TravisJ Sep 20 '18 at 16:44

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.


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