I have a low-cost clock radio that also has a "beep" alarm which can be used in lieu of the radio. It also has a poor excuse of a battery back-up system.
Why a "poor excuse"? Well, IT DOESNT SET OFF THE ALARM OR EVEN DISPLAY THE TIME if the AC is lost and all that's present is the batteries. It only
"backs up" the time so that it'll be correct when the power comes back on-or, at least, that's what it's SUPPOSED TO DO!
I have found that it's nowhere near the right time when the power comes back on-even when it was a 10 minute power cut WITH BRAND NEW BATTERIES!!! It's often 3 to 4 hours off when the power comes back on. Also, it doesnt remember the alarm settings AT ALL, reguardless of how long or short the power outage was! so even if the power cut out for 2 seconds at night, then came back on again, before the alarm was supposed to go off, it STILL wouldn't work! That kind of defeats the purpose of a battery back-up, right? It wouldnt do you any good in ANY situation-ever!
How about a clock where it's 100% functional on AC OR batteries? Everything would function the same either way, including the alarm. It shouldn't be that hard to implement, but I've never seen a clock that does this
Of course you could just get a battery-only clock, like I currently use, but the problem is it SUCKS DOWN BATTERIES! it only lasts about a month or 2, which is not very long for a clock, because I've had watch batteries last for YEARS before they went dead! I guess that's because my clock is just crap, but still, if the clock just had the batteries as a "safety-net" in case the power goes out, it'd last a heck of a lot longer.
Also, just as an added bonus, if you put rechargeable AA or AAA batteries in the clock (whatever it uses depending on design) and then used them as backup while normally using it with AC, the clock would keep those batteries charged. If you DONT have rechargeable batteries in it, and the batteries are low, the display should indicate that somehow, with a flashing battery icon or something.