The Heaviside Function

This will be a short guide on the Heaviside function. This function is simple and it has uses in probability (financial mathematics), statistics and in partial differential equations (PDEs).

One may view the Heaviside function as a special case of the indicator function.

Suppose that we have an independent variable (as time) being t. The Heaviside step function is defined as:

H(t) = \begin{cases}1 & \text{if } t \geq 0 \\ 0 & \text{if } t < 0 \end{cases}

If we apply horizontal shifts, we can generalize this Heaviside step function to be:

H(t - a)= \begin{cases}1 & \text{if } t \geq a \\ 0 & \text{if } t < a \end{cases}

Suppose a = 3, we would have:

H(t - 3)= \begin{cases}1 & \text{if } t \geq 3 \\ 0 & \text{if } t < 3 \end{cases}


This Heaviside step function is right continuous.

There are other versions and definitions of the Heaviside function which are not discussed here.

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