The step response of continuous-time systems

In this paper, we will prove that a causal, linear, continuous-time, time-invariant system is bounded if the step response is a function of bounded variation. We then present an example of a causal, linear, continuous-time, time-invariant, bounded operator that does not have an impulse response in the normal sense. The example system response to an impulse input signal is zero and the impulse response does not contain any impulse functions. Hence, the continuous-time operators have a more complex structure than our intuition has led us to believe.