# Definition:Heaviside Step Function

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## Contents

## 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.

## Sources

- 1965: Murray R. Spiegel:
*Theory and Problems of Laplace Transforms*... (previous) ... (next): Chapter $1$: The Laplace Transform: Some Special Functions: $\text {VII}$. The Unit Step function - 2009: William E. Boyce and Richard C. DiPrima:
*Elementary Differential Equations and Boundary Value Problems*(9th ed.): $\S 6.3$

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