The picture on the left shows a graph of
The red line shows the cross section x=0, while the green curve highlights the cross section y=0. Click and drag on the picture to rotate it; type "F" after clicking on the picture to view the cross sections without the surrounding surface.1
For this function,
Because fx(0,0) does not exist, f(x,y) is not differentiable at (0,0).
A function can fail to be differentiable at a point even if its partials exist at that point.
1 Why F? The applet on this page uses F to toggle the display of polygon Faces, which are the computer graphics objects which make up the displayed surface.