my fish has survived a day in this water and he is healthy
I know it's too late and I am probably not trying to be nice, but are you serious? You cycle a tank for half a day and then make conclusions about the betta's health after another day?
Here is a good place to start learning about cycling your tank:
http
/www.fishforums.net/index.php?/topic/277264-beginners-resource-center/
The ammonia produced by your betta will build up with every minute he spends in the tank and anything above zero level is poisonous and highly toxic. Since bettas are quite hardy fish, he will suffer slowly, he can start getting fuzzy stuff, rotting fins, laying on the bottom, difficulty breathing, etc...and that's if you are lucky and he doesn't die on you quicker than you think. All this can start happening fairly soon without a way to reverse the effects of uncycled tank and if you don't take actions, you won't be long coming back to the forum asking how to save your dying fish.
So read up the above link, cycle your tank one way or another. If you are keeping the fish in this uncycled tank, then do daily 80-90% water changes, in the mean time get yourself a liquid test kit(API is one of the best) and test for ammonia and nitrItes, if any are not 0, then do a large 80-90% water change every day the test shows any levels different than 0. Eventually your filter will grow enough beneficial bacteria to reduce the ammonia and nitrItes to 0. This can take from 4 to 8 weeks time. Then you can reduce the water change to a 50% once a week.
And in relation to your other question, can bettas survive in salt water, yes they can but do you want your fish to be surviving or to be living its life as he should? Did you catch him by the sea by any chance to be asking this question?