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Given are the two functions: $f(x,y)=\sqrt [ 3 ]{ { x }^{ 2 }+{ y }^{ 2 } } \quad f(0,0)=0\\ f(x,y)=\frac { x*y }{ \sqrt { { x }^{ 2 }+{ y }^{ 2 } } } $

Both functions are continous in (0,0) already checked this. For the partial derivatives in P(0,0) both function got the value 0 I'm now having struggles to proof that they are not differentiable (my guess) by using the Definition with a linear approximation f(x,y)=f(0,0)+L(x,y)+R(x,y) since L(x,y) becomes always zero. Thanks for help

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2 Answers 2

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Here's a hint: if both the partial derivatives exists and are continuous in $(0,0)$ then $f$ is differenciable and $f'$ is continuous at $(0,0)$.

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  • $\begingroup$ i know this theorem, but since both partial derivatives become 0 in P(0,0) i cant say if they are continious or not without e.g epsilon criteria or something like that. or is it enough to know they are zero hence discontinous? $\endgroup$
    – MasterPI
    Commented Jun 25, 2017 at 20:49
  • $\begingroup$ And what does it mean for my liner Map to be zero? $\endgroup$
    – MasterPI
    Commented Jun 25, 2017 at 21:00
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Hint: check path $y=x$ for both functions in definition of derivative using limit

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