Distributions (or
generalized functions) are objects that generalize the classical notion of functions in
mathematical analysis. Distributions make it possible to
differentiate functions whose derivatives do not exist in the classical sense. In particular, any
locally integrable function has a distributional derivative. Distributions are widely used in the theory of
partial differential equations, where it may be easier to establish the existence of distributional solutions than classical solutions, or appropriate classical solutions may not exist. Distributions are also important in
physics and
engineering where many problems naturally lead to differential equations whose solutions or initial conditions are distributions, such as the
Dirac delta function (which is historically called a "function" even though it is not considered a genuine function mathematically).