# 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.

### 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.

### Off

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

can be understood as:

*$f$ is***off**until time $c$.

or:

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

### On

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

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.

## 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 - 1978: Ronald N. Bracewell:
*The Fourier Transform and its Applications*(2nd ed.) ... (previous) ... (next): Frontispiece - 1978: Ronald N. Bracewell:
*The Fourier Transform and its Applications*(2nd ed.) ... (previous) ... (next): Chapter $4$: Notation for some useful Functions: Summary of special symbols: Table $4.1$ Special symbols - 1978: Ronald N. Bracewell:
*The Fourier Transform and its Applications*(2nd ed.) ... (previous) ... (next): Inside Back Cover - 2009: William E. Boyce and Richard C. DiPrima:
*Elementary Differential Equations and Boundary Value Problems*(9th ed.): $\S 6.3$

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