This simple but clever continuity tester sports two LEDs: RED LED = circuit resistance is < approx 10K, GREEN LED = circuit resistance <10Ω. It consumes approx 8mA from a 9V battery. When not operating, battery discharge current is immeasurable so no ON/OFF switch is required.
The unique feature that makes this feasible is the way it applies power when the leads are connected to a resistance. When current flows through D1, transistor Q1 turns on and powers the op amp circuit. When power is applied, the non-inverting input of U1A is biased at 40mV. The circuit under is biased at approx 4mA with the current flowing through LED D1. When the circuit under test is below 10Ω, the voltage developed is below 40mV so the op amp (wired as a comparator) outputs a positive signal and turns on the Green LED. E = I*R = 4mA * 10Ω = 40mV
Op amp selection
I chose the inexpensive LM358 dual op amp—these cost only about $0.33 each—the 2nd section is unused. A single supply (ground sensing) op amp is required. Note that the LM741 will not work. An interesting property of the PNP Darlington transistor input stage is that when it is reversed biased (@ +9V and Vcc = 0V), it conducts no current. In checking this feature, I increased the voltage to 28V and determined that it conducts no measurable current at this voltage as well.
Stretching the performance
I believe that the sensitivity can be increased to detect <1Ω, however I did not test for this condition. To do so, reduce the value of R1 & R2 to 220Ω so that the test circuit current (and LED current) is approx 18mA. Then reduce the bias voltage at the non-inverting input of U1A to 18mV by reducing R5 to 2.2K. It possibly may be reduced even further, but this may require an op amp with low input offset voltage such as the LM158 (Vos = 2mV Max). (The garden-variety LM358 has Vos = 7mV Max.) Protoboard photo
Note that I used a blue LED rather than a green one.