{They ask you (O Muhammad (sallallaahu alaihi wa sallam) about the new moons. Say: “These are signs to mark fixed periods of time for mankind…”} Surah 2 Verse 189

And the Prophet ﷺ said:

“Start observing Sawm (fasts) on seeing the crescent moon (of Ramadhaan), and give up observing saum (fast) on seeing the crescent-moon (of Shawwal).” Bukhaari

So fasting is linked to the sighting of the new moon, and giving up of fasting [end of Ramadan] is linked to the sighting of the new moon; and the month may be 30 days and it may be 29 days in accordance to the sighting of the new moon. So if it is not seen on the night of the 30th day [*translators note: bearing in mind the night precedes the day], then if this is relating to Sha’ban, the people arise [on the 30th day of Sha’ban] not fasting, because in this day is doubt [whether Ramadan has begun or not], and if this occured at the end of Ramadan then they fast the 30th day due to the origin of the affair being the establishment and continuity of Ramadan, just like in the first scenario the continuity of Sha’ban remains as the origin and default.