Definition:Heaviside Step Function

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Let $c \ge 0$ be a constant real number.

The Heaviside step function on $c$ is the real function $u_c: \R \to \R$ defined as:

$\map {u_c} t := \begin{cases} 1 & : t > c \\ 0 & : t < c \end{cases}$

If $c = 0$, the subscript is often omitted:

$\map u t := \begin{cases} 1 & : t > 0 \\ 0 & : t < 0 \end{cases}$

There is no universal convention for the value of $\map {u_c} c$.

However, since $u_c$ is piecewise continuous, the value of $u_c$ at $c$ is usually irrelevant.

Heaviside Step Function with Two Variables

Let $u: \R \to \R$ be the Heaviside step function.

Then the Heaviside step function of two variables is the real function $u : \R^2 \to \R$ defined as the product of two step functions of one variable:

$\map u {x, y} := \map u x \map u y$

In other words:

$\map u {x, y} := \begin{cases} 1 & : \paren {x > 0} \land \paren {y > 0} \\ 0 & : \text {otherwise} \end{cases}$

Graph of Heaviside Step Function

The graph of the Heaviside step function is illustrated below:


Off and On

Let $t$ be understood as time.

Let be $f$ a function of $t$ used to model some physical process.


$\map {u_c} t \map f t$

can be understood as:

$f$ is off until time $c$.


$f$ does not start until time $c$.


$\map {u_c} t \map f t$

can be understood as:

$f$ is on after time $c$.

Also denoted as

The Heaviside step function can also be denoted:

  • $\map {H_c} t$
  • $\map {\theta_c} t$

Variants of the letter $u$ can be found:

  • $\map {\UU_c} t$
  • $\map {\operatorname u_c} t$

Some sources bypass the need to use a subscript, and present it as:

$\map {\UU} {t - c} = \begin {cases} 1 & : t > c \\ 0 & : t < c \end {cases}$

Also known as

This is also called the unit step function.

Some sources merge the terminology and refer to it as Heaviside's unit function, or Heaviside's unit step.

Also see

  • Results about the Heaviside step function can be found here.

Source of Name

This entry was named for Oliver Heaviside.