Most modern DMMs as well as continuity testing can also measure capacitance, and this is what I use to find breaks. Having found which wire is broken using a continuity test I then measure the capacitance at one of the plugs from that wire to the screen (if there is no screen I connect all the other wires together). I then do the same at the other end of the cable. The ratio of the capacitance is then the ratio of the distances from the ends to where the break is.
With a bit of fiddling around:
Assume c1 is the capacitance at one end and d1 is its unknown distance. Similarly c2 is the capacitance from the other end and d2 the associated distance. dt is the known total cable length, and ct is the total capacitance, c1 + c2.
d1 / c1 = d2 / c2
but dt / ct = d1 / c1 = d2 / c2
d1 = dt x c1 / (c1 + c2)
Plonking in some figures for a 10m cable where we find we have capacitances of 100pF and 400pF
d1 = 10 x 100 / (100 + 400) = 2
So the break is 2m from one end. As a sanity check the break will be closest to the end with the lowest capacitance.
Because this relies entirely on ratios we don't need to know the cable's specific capacitance, and can work on any cable of any length provided we have a sensibly measurable capacitance.
It wasn't me!
(Well, actually, it probably was)