Unit Step Function

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

note: at 0, it's 1
You can express piecewise functions in terms of the Heaviside function

Example

$g (x) = {\begin{cases} 0, x < 0 \\ x (2 - x), 0 \leq x < 2 \\ 0, x \geq 2 \end{cases}$

Kinda self explanatory

\begin{aligned} g (x) & = 0 H (- x) + x (2 - x) (H (x) - H (x - 2)) + 0 (1 - H (2 - x)) \\ = x (2 - x) (H (x) - H (x - 2)) \\ = x (2 - x) H (x) - x (2 - x) H (x - 2) \end{aligned}

is a pulse, or
First term turns "on" parabola, second turns "off" parabola

harder

$\begin{aligned} f (x) & = {\begin{cases} 0, x < 0 \\ 2 x, 0 \leq x < 2 \\ 2, 2 \leq x < 3 \\ 6 - x, x \leq 3 \end{cases} \\ f (x) & = 2 x (H (x) - H (x - 2)) + 2 (H (x - 2) - H (x - 3)) + (6 - x) (H (x - 3)) \\ = 2 x H (x) + (2 - 2 x) H (x - 2) + (4 - x) H (x - 3) \end{aligned}$

you can go the other way too

Example

$f (x) = 4 x H (x) + (4 - 4 x) H (x - 1) + (12 - 4 x) H (x - 3) + (4 x - 16) H (x - 4)$
$x < 0, f (x) = 0$
$0 \leq x < 1, f (x) = 4 x$
$1 \leq x < 3, f (x) = 4 x + (4 - 4 x) = 4$
$3 \leq x < 4, f (x) = 4 + (12 - 4 x) = 16 - 4 x$
$4 \leq x, f (x) = 16 - 4 x + 4 x - 16 = 0$