Applet: Non-differentiable function with partial derivatives and no tangent plane
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Since the partial derivatives of this function $f(x,y)$ exist and are zero at the origin, the only possible candidate for a tangent plane is the horizontal plane $z=0$ shown here. Although this plane is tangent along the $x$ and $y$ directions, the slopes of the function clearly do not match in other directions. Therefore, there is no tangent plane at $\vc{a}=(0,0)$, and the function is not differentiable there. You can drag the blue point on the slider to remove the folds in the surface, but that does not change the partial derivatives at the origin. The wrinkle at the origin is enough to make the function non-differentiable.
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