simple ardunio question

can the Arduino be left on all the time?
I know it will run out of battery.
but will it stop working correctly if I leave it on for a long period of time?

Yes

As long as the code is carefully written, and all circuitry is correct, then there is no reason for it to ever need to be shut off.

I have one here that is monitoring all my snakes and also controlling the lighting.
It has currently been running for 97 days on this occasion, and apart from me changing the code or powercuts, I have never had to reset it.

so it can be left on for a long time

Yes

In short, yes, you can leave it on for a long time.
In many everyday devices, micro-controllers run all the time. Many times even when the device is off.