Non differentiable function with partial derivatives?

Question

Level curves of the considered function

Hello there,

I am having some trouble with the function defined by $f(x,y) = \frac{xy(x^2-y^2)}{x^2+y^2}$ whenever $(x,y) \ne (0,0)$ and $f(0,0) = 0$.

On one hand the level curves diagram suggests the function is not differentiable at the origin.

Also, I can compute the partial derivatives $f'_x$ and $f'_y$ with the normal rules away from the origin, and by the definition (with the limit) at $(0,0)$, both are $0$. The partial derivatives seem to be continous, since what one obtains is a rational function having an homogeneous numerator of degree 5, and a denominator of degree 4.

But one knows that having continuous partial derivatives implies differentiability...

Can someone help, please?

Answer 1

Hint: If $|f(x,y)| = o(\sqrt {x^2+y^2}),$ then $Df(0,0)=0.$

Answer 2

Here is an image of the surface as drawn by GeoGebra. It is actually a four part saddle point at (0,0,0):