The Importance of Continuity for Linear Time-Invariant Systems [Lecture Notes]

Linear time-invariant (LTI) systems play a fundamental role in signal processing. Continuity is an important property of LTI systems, without which many conclusions about LTI systems, such as convolution formula and commutative law, are not true in general. However, this concept does not receive as much attention as it should in the literature of signal processing. In this "Lecture Notes" article, we provide a detailed discussion on the importance of continuity for LTI systems. For simplicity, we mainly consider the discrete-time signals and systems.