The picture on the left shows a graph of
The red line shows the cross section x=0, while the green line 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,
Both of the partials exist at the origin, but the function clearly is not differentiable at (0,0); if the surface had a tangent plane there it would simultaneously have to be both z=x and z=-x.
Most calculus books include a theorem saying that if the partials are continuous at and around a point, a function is differentiable. The reason this example doesn't contradict the theorem is that the partials are not continuous at the origin.
However, it's also possible that a function is differentiable with discontinuous partials.
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.