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.

Off and On
Let $t$ be understood as time.

Let be $f$ a function of $t$ used to model some physical process.

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

 * Definition:Kronecker Delta


 * Definition:Dirac Delta Function