[Math] How to you rewrite piecewise functions in terms of the unit step $u(t-a)$

Question

Consider $
u(t-a) =
\begin{cases}
0, & \text{if }t<a \\
1, & \text{if }t\geq a
\end{cases}
$

How can we rewrite a function like $
f(t) =
\begin{cases}
\cos2t, & \text{if }0\leq t \lt 2\pi \\
0, & \text{if }t\geq 2\pi
\end{cases}
$ in terms of the unit step function? My textbook writes this particular example as $f(t) = [1-u(t-2\pi)]\cos2t$, but I don't understand how this was formulated nor how I can formulate other piecewise functions in terms of the unit step similarly.

Answer 1

Consider the function $f(a,b) = u(t-a)-u(t-b); a>b$. This function is 1 in $[a,b)$ and 0 elsewhere. So, suppose you want to write $$g(x) = \begin{cases} g_1(x), & \text{if } a_1<x<a_2 \\ g_2(x), & \text{if } a_2<x<a_3\\ \ldots \end{cases}$$

Then, $$g(x) = g_1(x)f(a_2,a_1) + g_2(x)f(a_3,a_2)+\ldots$$