the Heaviside step function
/ HEV-ee-side /
The Heaviside step is the simplest switch: it is 0 before a certain moment and 1 after it. Written H(x), it equals 0 for x < 0 and 1 for x > 0, with a single jump of size one at the origin. It models anything that turns on at an instant — a force applied at time zero, a voltage suddenly connected, the state of a system before and after an event. Unlike the delta, the Heaviside step really is an ordinary function; it is just discontinuous.
The reason it appears in distribution theory is its derivative. Classically H has no derivative at the jump, and its derivative is 0 on each side, so the ordinary slope tells you nothing about the jump. But as a distribution H can be differentiated, and the answer is exactly the Dirac delta: the distributional derivative of H is delta. The jump of size one becomes a unit spike of derivative right at the point of the jump. This is the cleanest illustration of how distributional differentiation sees what classical differentiation throws away — the size and location of a jump appear as a delta.
The Heaviside step matters as the bridge between jumps and point sources. It explains why integrating a delta gives a step, and why a fundamental solution often has a jump (the response of a system whose source switches on). In one stroke it shows that differentiation is not lost at a discontinuity when you work with distributions; it is enriched.
Differentiate H as a distribution: for any probe phi, the rule (H', phi) = -(H, phi') equals -(integral from 0 to infinity of phi'(x) dx) = -(phi(infinity) - phi(0)) = phi(0). That is exactly (delta, phi), so H' = delta.
The distributional derivative of a unit jump is a unit delta sitting exactly at the jump.
The value of H exactly at 0 (often taken as 1/2) does not matter for its action as a distribution: changing a function at a single point changes no pairing integral, so it is the same distribution either way.