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.png

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

 * Definition:Kronecker Delta


 * Definition:Dirac Delta Function