Can a betta fish live in purified water?

Yes I'd imagine so though I've not tried it, just use filtered, or boiled or water left to rest a day to remove the chlorine, though some water does have a lot of heavy metals that are not good for any fish. My experience has been they can live quite happily in many types of tanks from as small as a cup of water, they tend to be aggressive to many other fish in the same tank though and do need to be fed so keep in mind this will "muck up" the tank. Change at least 10% up to 25% some say no more than 50% but I think that's a bit drastic - so change a bit of the water in the tank each week to keep it clean.