If you are looking for a simple way to build a continuity circuit, there isn't anything much simpler than a battery and light bulb.
1st lead wire----battery---jumper wire---bulb----2nd lead wire
Connecting the leads to an unbroken conductor will cause the bulb to light, connecting to a broken conductor will fail to light the bulb. Good practice when doing continuity tests is
1) to make sure no secondary power source is in the circuit, as this could damage both tester and circuit
2) touch tester leads together prior to checking the conductor to make sure bulb and battery are working
3) touch leads together after checking conductor to make sure bulb and battery are still working.
edit...after doing a little thinking about your question, I believe you may be asking about checking continuity with a cable or wire that is connected in circuit and insulated where making electrical contact with a tester leads is difficult.
While I'm not aware of any non-contact type of continuity test, there ARE alternating current voltage testers that pick up the fields created by AC voltage that can be used to trace a wire to find the point of break. But these only work with AC, typically mains. Another way of determining continuity in a live circuit is with a clamp on amp-probe...if a wire that is supposed to be live and carrying current shows no amps, then the troubleshooting person could trace voltages to determine the point of break.