# Definition:Heaviside Step Function

## Definition

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. ## Also denoted as

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

Variants of the letter $u$ can be found:

• $\map {\mathcal U_c} t$
• $\map {\operatorname u_c} t$

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

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

## 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.

## Comment

If $t$ is understood as time and $f$ some function of time used to model some physical process, then:

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

is often understood as:

$f$ is off until time $c$ and then on after time $c$

or:

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

## Also see

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

## Source of Name

This entry was named for Oliver Heaviside.