tangent planes and linear approximations and partial derivatives

Question

I have to study tangent planes and linear approximations, there is this theorem :

THEOREM: if the partial derivatives $f_x$ and $f_y$ exist near $(a,b)$ and are continuous at $(a,b)$, then $f$ is differentiable at $(a,b)$

Actually, it's foggy in my head about what I have to do to show their existences, because everytime I just calculating the partial derivatives straightforward.

Could you, please, give to me a short example where one of the partial derivative doesn't exist

score 0 · Accepted Answer · answered Oct 26 '13 at 16:34

0

How about $f(x,y) = |x|y$? Or, for an example where the partial derivatives exist everywhere except at a single point, the cone $f(x,y) = \sqrt{x^2+y^2}$.

answered Oct 26 '13 at 16:34

user7530

49,280

I'm right when I say both exist near $(0,0)$ but they are not continuous around $(0,0)$ ? – The Unholy Metal Machine Oct 26 '13 at 16:55

tangent planes and linear approximations and partial derivatives

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