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Heaviside Step Function
The Heaviside step function, using the half-maximum convention
The Heaviside step function, or the unit step function, usually denoted by H or ? (but sometimes u, 1 or 1), is a discontinuousfunction, named after Oliver Heaviside (1850-1925), whose value is zero for negative arguments and one for positive arguments. It is an example of the general class of step functions, all of which can be represented as linear combinations of translations of this one.
The function was originally developed in operational calculus for the solution of differential equations, where it represents a signal that switches on at a specified time and stays switched on indefinitely. Oliver Heaviside, who developed the operational calculus as a tool in the analysis of telegraphic communications, represented the function as 1.
The simplest definition of the Heaviside function is as the derivative of the ramp function:
In operational calculus, useful answers seldom depend on which value is used for H(0), since H is mostly used as a distribution. However, the choice may have some important consequences in functional analysis and game theory, where more general forms of continuity are considered. Some common choices can be seen below.
where a larger k corresponds to a sharper transition at x = 0. If we take H(0) = , equality holds in the limit:
There are many other smooth, analytic approximations to the step function. Among the possibilities are:
These limits hold pointwise and in the sense of distributions. In general, however, pointwise convergence need not imply distributional convergence, and vice versa distributional convergence need not imply pointwise convergence. (However, if all members of a pointwise convergent sequence of functions are uniformly bounded by some "nice" function, then convergence holds in the sense of distributions too.)
Often an integral representation of the Heaviside step function is useful:
where the second representation is easy to deduce from the first, given that the step function is real and thus is its own complex conjugate.
Since H is usually used in integration, and the value of a function at a single point does not affect its integral, it rarely matters what particular value is chosen of H(0). Indeed when H is considered as a distribution or an element of L? (see Lp space) it does not even make sense to talk of a value at zero, since such objects are only defined almost everywhere. If using some analytic approximation (as in the examples above) then often whatever happens to be the relevant limit at zero is used.
There exist various reasons for choosing a particular value.
H(0) = is often used since the graph then has rotational symmetry; put another way, H - is then an odd function. In this case the following relation with the sign function holds for all x:
H(0) = 0 is used when H needs to be left-continuous. In this case H is an indicator function of an open semi-infinite interval:
In functional-analysis contexts from optimization and game theory, it is often useful to define the Heaviside function as a set-valued function to preserve the continuity of the limiting functions and ensure the existence of certain solutions. In these cases, the Heaviside function returns a whole interval of possible solutions, H(0) = [0,1].