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The Heaviside step function, denoted as H(x) or θ(x), is a mathematical function that is zero for negative arguments and one for positive arguments. At zero, its value can be defined as 0, 1/2, or 1, or it can be left undefined. This discontinuity at x=0 is a defining characteristic of the function.

In signal processing and control theory, the Heaviside step function is fundamental. It represents an instantaneous switch or a signal that turns on at a specific time. For example, it can model the sudden application of a voltage or current. Its integral is the ramp function, and its derivative is the Dirac delta function, making it a crucial element in the study of differential equations and distributions.

The Heaviside step function can be expressed mathematically as:

H(x) = { 0 if x < 0, 1 if x >= 0 } (with variations at x=0).

It is also closely related to the Kronecker delta function, which is similar but defined only for discrete values. The Heaviside step function's ability to represent abrupt changes makes it indispensable in various scientific and engineering disciplines.