In a 1g tank, I'm assuming you don't have a filter on the tank. If thats the case, then you should change out 25% of the water every few days to keep it clean and prevent fin rot. If thats the betta's tail in your avatar that you are talking about, it doesn't look like fin rot to me right now. Fin rot is usually caused by poor water quality and/or stress.
Keep the betta's water clean by changing the water out every few days (dechlorinated water). Try to match the new water temp with the tank water.
Do you have a heater? If not, what temperature does the tank stay at?
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