have a weak derivative because its distributional derivative, namely the Cantor distribution, is a singular measure and therefore cannot be represented by a function. Properties If two functions are weak derivatives of the same function, they are equal except on a set with Lebesgue measure zero, i.e., they are equal almost everywhere. If we consider equivalence classes of functions such that two functions are equivalent if they are equal almost everywhere, then the weak derivative is unique. Also, if u is differentiable in the conventional sense then its weak derivative is identical (in the sense given above) to its conventional (strong) derivative. Thus the weak derivative is a generalization of the strong one. Furthermore, the classical rules for derivatives of sums and products of functions also hold for the weak derivative. Extensions This concept gives rise to the definition of weak solutions in Sobolev spaces, which are useful for problems of differential equations and in functional analysis. See also Subderivative Weyl's lemma (Laplace equation) References Generalized functions Functional analysis Generalizations of the derivative Generalizations