According to the confuser if the anti-collision light fails (I have no idea how you would tell...) you can carry on in day time.
Yep, that's the type of question they will ask, if x light fails, can you continue. You won't be asked how you would identify the failure.
In reality, when you switch electrics on and off you will likely see a jump on the ammeter. This is a good indication if things like pitot heat are working, and when strobes are on the needle will often flicker up and down. This doesn't work if only one light has failed, you will still get a jump as the other lights go on, but it's better than nothing.
Also, in a C172 the Nav Lights are positioned such that at night you can see they are on. This may be true for other aircraft too.