Test with the multimeter. The multimeter will beep if there are no breaks in the wire. Use a sharp knife to cut the insulation from the wire, following the instructions below. Be careful not to cut the wire inside.
Make a slit next to the plug, and one next to the earpiece.
The bare copper wire usually has a clear protective coat. Gently scrape it away with a knife.
Touch the wire in one slit with the black multimeter lead. Touch the other slit with the red lead. If it beeps, the problem is in the plug or earpiece.
If it does not beep, make a slit halfway along the cable and test each half of the cable.
Make another cut in the half that does not beep. Repeat until you've found two points a few inches (several cm) apart that do not cause the multimeter to beep.
Continue on to Fixing the Cable, skipping the test step.