the Dirac delta function
/ dee-RAK /
Imagine all of a unit of something -- mass, charge, probability -- squeezed onto a single point: infinitely tall, infinitely thin, yet with total amount exactly one. That idealization is the Dirac delta. It is not a function in the ordinary sense (no honest function is zero everywhere but has area one); it is a limit -- the sharpening of an ever-narrower, ever-taller spike.
The Dirac delta delta(x) is defined not by its values but by what it does inside an integral: it is the object for which the integral of delta(x) dx = 1 and, crucially, the sifting property, the integral of f(x) delta(x - a) dx = f(a) -- it plucks out the value of any test function at the point a. Rigorously it is a distribution (a generalized function), a continuous linear functional on smooth test functions, and it is the limit of nascent deltas such as a narrowing Gaussian, or the value (1/(2 pi)) times the integral of e^(i k x) dk. Its derivative is defined by moving the derivative onto the test function: the integral of delta'(x) f(x) dx = -f'(0).
It is the physicist's tool for anything point-like or instantaneous: a point charge's density is q delta(r - r0), a sharp impulse is F delta(t), and the orthonormality of continuous bases (position, momentum) reads the inner product of x and x' = delta(x - x'). It is the source term that defines Green's functions (L G = delta), and its Fourier representation is what makes the transform of a pure exponential a delta spike -- the mathematical statement of a perfectly sharp frequency.
The sifting property in action: the integral from -infinity to infinity of cos(x) delta(x - pi) dx = cos(pi) = -1. The spike at x = pi ignores cos(x) everywhere else and reports only its value there.
The delta function samples a function at a single point.
It is not a function, and manipulating it as one invites nonsense (delta(0) is not a number, and delta(x)^2 is undefined). It is legitimate only inside an integral against a smooth test function -- that is the only meaning it has.