Continuity testers are important measurement devices. As the name suggests, you can test if traces on a PCB are broken or not. You can find short circuits and if there is one, the tester makes a sound. Building these small devices is pretty easy.
During my vocational training I built some of them, but this one is the best of them all since you can test whether or not there is AC on the wires/conductors. E.g. you are testing a transformer on the primary coil and it still is turned on, this tester gives you a warning tone and a warning light.
And all this is built quite simple.
A 9V battery is powering everything. A pnp-transistor with a resistor between its base and emitter is powering an astable multivibrator running at ~1kHz. This signal is then passed on to npn-transistor driving a speaker. The multivibrator only is running if the two connectors for the probes are shorted out (or have a resistance of ~1kOhm max).
If however mains power is connected to the probe leads the multivibrator is turned on and off in 50Hz (here in Europe). This results in a humming tone. Also a small glow lamp is connected via 1.8MOhm.
Everything is compact enough to fit in a BOPLA E-420 casing. The glow lamp got a small red window and the speaker is glued to the inside part of the top cover. It also has some holes for the sound to pass through.
Here are schematic and a picture: