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 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.

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