 Distribution theory

Distribution theory extends ordinary functions by letting them act on smooth test functions. A locally integrable function $f$ acts by $⟨f,\varphi ⟩=\int f\varphi$, while the Dirac delta distribution acts by $⟨{\delta }_a,\varphi ⟩=\varphi (a)$. The latter is a point source, rather than an ordinary function with a very large value. The space of test functions and its topology make distributions continuous linear functionals.^[ref]

Integration by parts supplies a distributional derivative even when ordinary differentiability fails:For the Heaviside function $H$, $⟨H^{\prime },\varphi ⟩=-{\int }_0^{\infty }{\varphi }^{\prime }(x)dx=\varphi (0)$, so $H^{\prime }={\delta }_0$. A jump has become a point source under differentiation. This is useful for impulses in differential equations, concentrated loads and interface jump conditions.

The extension has limits: multiplying a distribution by a smooth function is always defined, but arbitrary products of distributions are not. The theory controls which operations are meaningful through their action on test functions, rather than by manipulating symbolic point values.