basic oscilloscope question

The single pulse would be very narrow. Ideally, it contains all frequencies. A signal generator typically creates a constant output waveform at a constant frequency. Some only output sine waves. Others might ouput square waves, triangle waves, etc; some with varying duty cycles.

I think of a signal generator as a device that produces sine waves of selectable frequency and amplitude.

A Function Generator is a more versatile device - it will produce square, triangle and pulse signals of selectable frequency and amplitude (and pulse width or duty cycle for square waves and pulses).