Heaviside function

The Heaviside function, often written as "H"(x), is a non-continuous function whose value is zero for a negative input and one for a positive input.
The function is used in the mathematics of control theory to represent a signal that switches on at a specified time, and which stays switched on indefinitely. It was named after the Englishman Oliver Heaviside.
The Heaviside function is the integral of the Dirac delta function: "H"′(x) = "δ"(x). This is sometimes written as
Discrete form.
We can also define an alternative form of the Heaviside step function as a function of a discrete variable "n":
where "n" is an integer.
Or
The discrete-time unit impulse is the first difference of the discrete-time step
This function is the cumulative summation of the Kronecker delta:
where
is the discrete unit impulse function.
Representations.
Often an integral representation of the Heaviside step function is useful:
"H"(0).
The value of the function at 0 can be defined as "H"(0) = 0, "H"(0) = ½ or "H"(0) = 1. In particular: