Heaviside Function

AKA Unit Step Function

VERY IMPORTANT
- Prof. JINTAO Deng

Perhaps the simplest piecewise-defined function, and the most important one.

Definition

left = -3; right = 3;
bottom = -1; top = 2;
---
y = 0 \{x < 0\}
y = 1 \{x \ge 0\}
(0, 0) | OPEN
(0, 1) | POINT

Multiplication

Shifting

Example -

left = -2; right = 4;
bottom = -1; top = 5;
---
y = 0 \{x < 1\}
(1, 0) | OPEN
y = x^2 \{x \ge 1\}
(1, 1) | POINT

Converting from Heaviside to Piecewise

Write in Piecewise-Defined Form

Process: start with the "lowest" interval (in this case, ), and work your way up, compounding the previous heaviside function's results on top

Write in Heaviside Form

We have

At , we need to replace with . We can do this by adding to remove the , and then adding

At , we need to replace with , so we subtract and add