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. |