let f(t) be an elementary function defined for every real number t

denote the unit step function as U(t-a), t >= 0, a real

such that U(t-a) = {0 if 0<=t<a, and 1 if t>= a

then

f(t)U(t-a) = {0 if 0<=t<a, and f(t) if t >= a

No problem.

What happens when we compute the product U(t-a)p(t), where p(t) is itself a piecewise function? i.e. p(t) = { condition 1, 2, ..., n