Expressing a Piecewise Function with Shifted Unit Step Functions

Topics: Piecewise Function

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(theorem)

Let $f(t)$ be a piecewise function of the following form:

$$f(t)=\{\begin{array}{ll}g(t)& \text{if }0\le t<a\\ h(t)& \text{if }a\le t<b\\ k(t)& \text{if }b\le t\end{array}$$

We can write such a function as follows, by using the shifted unit step function $U$:

$$\begin{array}{rl}f(t)& =g(t)-g(t)U(t-a)+h(t)U(t-a)-h(t)U(t-b)+k(t)U(t-b)\\ & =g(t)+[h(t)-g(t)]U(t-a)+[k(t)-h(t)]U(t-b)\end{array}$$