Continuous-time linear systems: folklore and fact

We consider a family of continuous input-output maps representing linear time-invariant systems that take a set of signals into itself. It is shown that this family contains maps whose impulse response is the zero function, but which take certain inputs into nonzero outputs. It is shown also that this family contains members whose input-output properties are not described by their frequency-domain response functions, and that the maps considered need not even commute.