Differentiability of 2-Variable Functions
 |  |
 |  |
| Several views of the graph of f(x,y) = 4x2 - y2 + 2y
near the point (-1,2,4), shown in red. Notice that as we zoom in toward (-1,2,4) the graph starts to look more and more like that of its tangent plane. This is because f is differentiable at this point. | |
 |  |
 |  |
| Several views of the graph of f(x,y) = x2y/(x2 + y2)
near the point (0,0,0), shown in red. Notice that as we zoom in toward (0,0,0) the "crinkle" in the graph persists. This is because f is not differentiable at this point (even though both fx(0,0) and fy(0,0) exist!). | |