A (Schwartz) distribution is a generalized function, such as the Dirac delta function δ‎. Unlike classical functions, distributions are not defined at points but by integrals. A test function φ‎:ℝ→ℝ is a smooth (i.e. infinitely differentiable) function such that φ‎ is zero outside a bounded interval. If f:ℝ→ℝ is a continuous function, then the integral

exists. Further, knowing all such integrals, it is possible to determine f entirely. A distribution is a linear functional on the space of test functions (which is continuous in a technical sense); the distributions generalize (locally integrable) functions by means of the previous integrals. As a distribution, 〈δ‎,φ‎〉 = φ‎(0), as expected from the sifting property. A distribution F is differentiable by the rule 〈F’,φ‎〉 = –〈F,φ‎’〉 (compare with integration by parts). As a distribution, the delta function is the derivative of the Heaviside function, even though the latter is discontinuous as a function.