Heaviside Step Function — Definition, Formula & Examples

The Heaviside step function is a piecewise function that equals 0 for negative inputs and 1 for positive inputs. It acts like a mathematical "on/off switch" that jumps from 0 to 1 at a specified point.

The Heaviside step function $H(t)$ is defined as $H(t) = 0$ for $t < 0$ and $H(t) = 1$ for $t \geq 0$ . A shifted version $H(t - c)$ transitions from 0 to 1 at $t = c$ . The value at exactly $t = 0$ is sometimes defined as $\tfrac{1}{2}$ depending on the convention used.

Key Formula

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

Where:

$t$ = The input variable (often representing time)
$H(t)$ = Output: 0 before the jump, 1 at and after the jump

How It Works

The Heaviside function lets you "turn on" another function at a chosen time. To activate a function

f(t)

starting at

t = c

, multiply it by

H(t - c)

. For example,

\sin(t) \cdot H(t - 3)

equals zero when

t < 3

and equals

\sin(t)

when

t \geq 3

. In Laplace transforms, the shifted Heaviside function has the transform

\mathcal{L}\{H(t-c)\} = \frac{e^{-cs}}{s}

, which makes it essential for solving differential equations with discontinuous forcing functions.

Worked Example

Problem: Write a single expression for the function that equals 0 for

t < 2

, equals 5 for

2 \leq t < 6

, and equals 0 again for

t \geq 6

Step 1: Turn on the value 5 at

t = 2

using a shifted Heaviside function.

5\,H(t - 2)

Step 2: Turn off the value 5 at

t = 6

by subtracting another shifted Heaviside function.

5\,H(t - 2) - 5\,H(t - 6)

Step 3: Verify: for

t < 2

, both terms are 0. For

2 \leq t < 6

, the first term gives 5 and the second is still 0. For

t \geq 6

, both terms are 5, so the difference is 0.

f(t) = 5\bigl[H(t-2) - H(t-6)\bigr]

Answer:

f(t) = 5\,H(t-2) - 5\,H(t-6)

, which creates a rectangular pulse of height 5 on the interval

[2, 6)

Why It Matters

The Heaviside function is indispensable in differential equations courses when modeling systems subjected to sudden forces — a circuit switched on at a specific time, or a load suddenly applied to a beam. It pairs directly with Laplace transforms, giving you a systematic algebraic method to solve problems that would be cumbersome with purely piecewise techniques.

Common Mistakes

Mistake: Confusing

H(t - c)

with

H(c - t)

, which reverses the on/off behavior.

Correction:

H(t - c)

turns on at

t = c

(equals 0 before, 1 after).

H(c - t)

does the opposite: it equals 1 before

t = c

and 0 after. Pay attention to the sign inside the argument.