The continuity function is designed to give an audible indication of resistance that is less than some threshold value.
To be useful, it will be designed to respond much more quickly than the display so that a tech can quickly 'buzz out' wiring and such like without waiting for the reading to settle or even taking his or her eyes off the test probes. It's specifically designed so that the voltage does not turn diodes on, it won't respond to resistors above a certain value and so on, so that it (usually) responds to just a fairly solid electrical connection.
The values will vary somewhat by manufacturer, but here is an excerpt from the Fluke 177 DMM manual:
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As you can see it has hysteresis and a pulse stretcher that allows brief breaks to be detected. This is done by circuitry that is mostly operating in parallel to the main ADC function. Some crummy cheap meters have a continuity beep that is dependent on waiting for an ADC result, but they are not very useful. Avoid!
