The picture on the left shows a graph of The red curve 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} This function is differentiable at (0,0), and both f_{x}(0,0) and f_{y}(0,0) equal 0. These partials exist for essentially the same reason that x^{2}sin(1/x) is differentiable in Calculus 1. In fact, it's exactly the same reason: If x=0, then f(0,y)=y^{2}sin(1/|y|) Most calculus books include a theorem saying that if the partials are continuous at and around a point, a function is differentiable. This example shows that a function can still be differentiable even if its partials are not continuous. (You can verify this on your own, but both partials include a term that oscillates wildly as (x,y) approaches the origin.) ^{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. |