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 positive arguments. Different conventions concerning the value H(0) are in use. 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. Heaviside developed the operational calculus as a tool in the analysis of telegraphic communications and represented the function as 1.

Taking the convention that H(0) = 1, the Heaviside function may be defined as:

For the alternative convention that H(0) = ⁠1/2⁠, it may be expressed as:

Other definitions which are undefined at H(0) include:

The Dirac delta function is the weak 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.)

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.

For a smooth approximation to the step function, one can use the logistic function:$${\displaystyle H(x)\approx {\tfrac {1}{2}}+{\tfrac {1}{2}}\tanh kx={\frac {1}{1+e^{-2kx}}},}$$where a larger k corresponds to a sharper transition at x = 0.