generalized function
The Dirac delta is supposed to be zero everywhere except at one point, infinite there, yet integrate to one — no honest function behaves like that. Rather than ban it, mathematicians widened the definition of 'function' to make sense of it. A generalized function, or distribution, is the result: an object you are not allowed to evaluate at a point, but which you can pair with smooth test functions to get a number. It is defined entirely by what it does inside an integral.
The precise idea is to stop asking 'what is the value of delta at x?' and instead ask 'what is the integral of delta(x) phi(x) dx?' for every smooth, rapidly decaying test function phi. The delta is defined by the rule that this integral equals phi(0) — it samples a test function at the origin. An ordinary function g also defines a distribution, via the map phi maps to the integral of g phi, and from this viewpoint every locally integrable function is already a distribution. The win is that distributions can always be differentiated: you define the derivative of a distribution by moving the derivative onto the test function (with a sign), so the Heaviside step has a derivative — and it is exactly the delta. Differentiation, which classically fails at jumps, never fails for distributions.
This is the rigorous home for everything in this field that relies on point sources. L G = delta is an equation between distributions; the jump condition is what that distributional equation forces on an otherwise classical G; the fundamental solution is a distribution. Distributions also justify the Fourier transform of a constant, of a pure sine, and of the delta itself, which is why they are indispensable in signal processing, quantum field theory, and PDE. The standing caveat, repeated because it is so often abused: the delta is not a function, you cannot square it or evaluate delta(0), and any manipulation must ultimately make sense under an integral against a test function.
The Heaviside step H(x) (0 for x < 0, 1 for x > 0) has no classical derivative at 0. As a distribution its derivative is defined by moving d/dx onto the test function: the integral of H phi' equals -phi(0), which is exactly the rule for -(-delta), so H' = delta.
Differentiating the step in the distributional sense yields the delta — the cleanest illustration of why generalized functions exist.
You cannot multiply two distributions in general — the product of two deltas, or delta squared, is undefined. This is not a technicality you may ignore: it is the deep reason naive nonlinear manipulations with deltas (and the renormalization headaches of quantum field theory) go wrong.