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Heaviside function

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File:Dirac distribution CDF.svg
The Heaviside step function, using the half-maximum convention

The Heaviside step function, H, also called the unit step function, is a discontinuous function whose value is zero for negative argument and one for positive argument. It seldom matters what value is used for H(0), since H is mostly used as a distribution. Some common choices can be seen below.

The function is used in the mathematics of control theory and signal processing to represent a signal that switches on at a specified time and stays switched on indefinitely. It was named in honor of the English polymath Oliver Heaviside.

It is the cumulative distribution function of a random variable which is almost surely 0. (See constant random variable.)

The Heaviside function is an antiderivative of the Dirac delta function: H′ = δ. This is sometimes written as

 H(x) = \int_{-\infty}^x { \delta(t)} \mathrm{d}t

although this expansion may not hold (or even make sense) for x = 0, depending on which formalism one uses to give meaning to integrals involving δ.

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