A sufficient condition for the differentiability of a function of two variables is that both partial derivatives exist and are continuous in a neighborhood of the point. This means that if these conditions are met, the derivative \mathbf{D}f exists. However, there are exceptions, as illustrated by the function f(x) = x² for rational x and -x² for irrational x, which is differentiable at 0 but nowhere else. This example highlights that a function can be differentiable at a point even if its partial derivatives are not continuous elsewhere. Thus, while the existence and continuity of partial derivatives are strong indicators, they do not guarantee differentiability in all cases.