What will be the amplitude modulated signal, if the input is an unit step signal? You can take any carrier signal (though I would recommend sine wave) and take any carrier frequency because that is not of interest here as I want to know the shape of the net modulated signal.
Before the start of the unit step, the waveform will be a constant amplitude sine wave at the carrier frequency. After the unit step, if we assume the amplitude of the step is sufficient for 100% modulation, the waveform will be a constant amplitude sine wave at twice the amplitude that existed before the start of the unit step. A unit step amounts to a DC level so that it affects the amplitude of the carrier but not its shape.
Before the step, the modulation signal is 0, so the output is 0. After the step, the modulation signal is 1, so the output will be a full amplitude carrier. This is the same as turning the carrier on from having been off.
Such modulation is called OOK (On/off keying), and is a common way to send morse code. The carrier is on for the dits and dahs, and off during the spaces in between them.