Electronic – Digital Multimeter “Continuity test” only one single beep

Sharp tips are able to poke through insulation, whether it be wire insulation, or the coatings on circuit boards. They are also handy if you need to stick it down into a connector between the wire and the body of the connector. Also, as parts get smaller, you need sharp tips to be able to contact one pin of an I.C. without shorting to the others. Lastly, a sharp point isn't as likely to slip when you look away to read the meter. The flow of current from a point isn't meaningful here because that's only relevant when there's an air gap present.

The response time of the continuity "beep" is important. If you want to test a number of contacts--especially if you're hunting for that one signal in a large set--you want to be able to do it fast. Otherwise, you have to touch the wire, then wait, then move on. It's very unsettling when doing continuity checks to have to rely on the notion that "it didn't beep, therefore there's no continuity". Did I wait long enough? Did I actually contact the wire? This is why you start by touching the meter leads together to be sure the beeper is actually working.

I'll assume you don't have or can't find a schematic for it (if you do let us know) so:

First thing is to open it up and take a look around the input circuitry, something blown/shorted may be easy to spot visibly (as Jim mentions any protection components are a good bet)
If the rest of the functions are still working, it may be just the continuity circuit which is damaged - if this is the only function which requires the buzzer on the meter, tracing back from the buzzer to any driving circuitry may prove helpful.