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:


 * $u_c \left({t}\right) = \begin{cases}

1 & : t > c \\ 0 & : t < c \end{cases}$

If $c = 0$, the subscript is often omitted:


 * $u \left({t}\right) = \begin{cases}

1 & : t > 0 \\ 0 & : t < 0 \end{cases}$

There is no universal convention for the value of $u_c \left({c}\right)$.

However, since $u_c$ is piecewise continuous, the value of $u_c$ at $c$ is usually irrelevant.

Also denoted as

 * $H_c \left({t}\right)$


 * $\theta_c \left({t}\right)$

Variants of the letter $u$ can be found:


 * $\mathcal U_c \left({t}\right)$


 * $\operatorname u_c \left({t}\right)$

Also known as
This is also called the unit step function.

Comment
If $t$ is understood as time and $f$ some function of time used to model some physical process, then:


 * $u_c \left({t}\right) f \left({t}\right)$

is often understood as:


 * $f$ is off until time $c$ and then on after time $c$

or:


 * $f$ doesn't start until time $c$.

Also see

 * Definition:Kronecker Delta