1. The Gemara relates Levi's odd Yom Kippur in Bavel due to an extended month.
2. Rava would commonly fast for two days and nights on Yom Kippur.
3. Rav Nachman would not usually fast for two days on Yom Kippur.
4. Witnesses should transgress Shabbos in order to testify about the new moon for two months: Nisan and Tishrei.
5. When the Beis ha'Mikdash stood, witnesses would transgress Shabbos to testify about any month.
A BIT MORE
1. Levi arrived from Eretz Yisrael knowing that the Beis Din there had extended Elul by one day. When he arrived in Bavel on Yom Kippur (the tenth of Tishrei), he realized that the Jews of Bavel had not received the message and had observed Yom Kippur the previous day.
2. This was in order to avoid eating on the real Yom Kippur in case he did not hear that Beis Din in Eretz Yisrael had extended Elul.
3. He did not deem his health good enough to do so, and was, therefore, very distressed one year when a person came to him after Yom Kippur and informed him that the Beis Din in Eretz Yisrael had extended Elul.
4. This is because it is extremely important that everyone know when the Yamim Tovim occur.
5. This was in order to ensure that the special Korban for Rosh Chodesh would indeed be brought on Rosh Chodesh.