Dirac delta function

How do you model an idealised instant — a hammer blow that delivers all its punch in zero time, a point mass with no size, a perfect spark of charge at a single point? You want something that is zero everywhere except one point, yet whose total effect is exactly 1. No ordinary function can do this, so physicists introduced a new kind of object: the Dirac delta, written delta(x), the idealised infinitely tall, infinitely narrow spike of unit area.

Honestly stated, delta(x) is not a function at all — it is a generalised function, or distribution. Its meaning lives entirely in how it acts inside an integral: the integral over all x of delta(x) f(x) dx = f(0), and more generally the integral of delta(x minus a) f(x) dx = f(a). That sifting property — the delta reaches in and plucks out the value of f at one point — is its complete definition. You may picture it as the limit of ever-narrower, ever-taller bumps (a tall thin Gaussian, say) each enclosing area 1, but the rigorous object is defined by what it does to test functions, not by any pointwise values. Its formal derivative behaves so that the integral of delta-prime(x) f(x) dx = minus f-prime(0).

The delta is the unit impulse of engineering: feed it into a system and the output is the impulse response, which then determines the response to any input by convolution. It is the point source of a Green's function, the point charge of electrostatics, the perfect sampling operation in signal processing, and the canonical right-hand side that builds fundamental solutions of the heat, wave and Laplace equations. The Heaviside step function is its integral, and differentiating a jump discontinuity produces a delta.