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\begin{align} f(t) = \begin{cases} 3t &\mbox{if } t \leq 3 \\ 12 & \mbox{if } 3<t\leq 7 \\ 0 & \mbox{if } t\geq 7 \end{cases} \end{align}

I'm confused as to how I can write this in terms of the unit step function with the first f(t) has $\leq 3$ instead of just $\lt 3$. Any help with this problem?

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I edited your latex, please check if this is still consistent –  sonystarmap Dec 14 '12 at 11:15
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This is besides the question, but your function is slightly flawed. At $t=7$, is $f(t)$ equal to 12 or 0? –  Paresh Dec 14 '12 at 11:34

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up vote 1 down vote accepted

Depends on the definition of the Heaviside function - it can be either 0, 1 or 0.5 at H(0).

If you take it to be zero, I think the following works:

[3tH(3-t) + 12 H(7-t) H(t-3)]H(7-t)

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Indeed, wolframalpha.com/input/… –  lhf Dec 14 '12 at 12:23

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