The unit step function, often denoted as $u(t)$, is a piecewise function that is zero for negative values of time and one for positive values, effectively 'turning on' at $t=0$. It is crucial in engineering and mathematics for modeling sudden changes in systems, such as a switch being turned on or off. The unit step function serves as a foundation for understanding discontinuous forcing functions and plays a significant role in the context of Laplace transforms, which are used to analyze linear time-invariant systems.
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