Heaviside step function

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Heaviside step
The Heaviside step function, using the half-maximum convention
General information
General definition	$${\displaystyle H(x):={\begin{cases}1,&x\geq 0\\0,&x<0\end{cases}}}$$
Fields of application	Operational calculus

The Heaviside step function, or the unit step function, usually denoted by H or θ (but sometimes u, 1 or 𝟙), is a step function named after Oliver Heaviside, the value of which is zero for negative arguments and one for nonnegative arguments.^[1] 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 Heaviside function may be defined as:

• a piecewise function:
  $${\displaystyle H(x):={\begin{cases}1,&x\geq 0\\0,&x<0\end{cases}}}$$

  
• using the Iverson bracket notation:
  $${\displaystyle H(x):=[x\geq 0]}$$

  
• an indicator function:
  $${\displaystyle H(x):=\mathbf {1} _{x\geq 0}=\mathbf {1} _{\mathbb {R} _{+}}(x)}$$

  
• the derivative of the ramp function:
  $${\displaystyle H(x):={\frac {d}{dx}}\max\{x,0\}\quad {\mbox{for }}x\neq 0}$$

  

The Dirac delta function is the derivative of the Heaviside function

$${\displaystyle \delta (x)={\frac {d}{dx}}H(x)}$$

Hence the Heaviside function can be considered to be the integral of the Dirac delta function. This is sometimes written as

$${\displaystyle H(x):=\int _{-\infty }^{x}\delta (s)\,ds}$$

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. In this context, the Heaviside function is the cumulative distribution function of a random variable which is almost surely 0. (See constant random variable.)

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.

Approximations to the Heaviside step function are of use in biochemistry and neuroscience, where logistic approximations of step functions (such as the Hill and the Michaelis–Menten equations) may be used to approximate binary cellular switches in response to chemical signals.