Rabbi Zeira said: When I was in Babylonia, I said with regard to that which was taught in a baraita: If Yom Kippur occurred on Shabbat eve, they would not sound the shofar as they did every Friday to herald the start of Shabbat; and if Yom Kippur occurred at the conclusion of Shabbat, they would not recite havdala to mark the end of the sanctity of Shabbat and the start of the sanctity of Yom Kippur, is a statement accepted by all.
When I went to Eretz Yisrael, I found Yehuda, son of Rabbi Shimon ben Pazi, who sat and said: This baraita is in accordance with the opinion of Rabbi Akiva, as it equates the sanctity of Yom Kippur with that of Shabbat. As, if you say that it is in accordance with the opinion of Rabbi Yishmael, then, since Rabbi Yishmael said that fats from Shabbat are offered on Yom Kippur, let them sound the shofar so that the priests will know that the fats from Shabbat are offered on Yom Kippur and they may begin offering them (Rav Hai Gaon). And I said to him: You cannot prove this from here, because priests are vigilant and can be trusted to know this on their own, and there is no need to sound the shofar.
There are two basic approaches to interpreting the question about blowing the shofar that was raised by Yehudah, son of Rabbi Shimon ben Pazi.
In the first approach, Rashi explains that the Gemara is not referring to shofar blasts between Shabbat and Yom Kippur, because the shofar was never sounded at the conclusion of Shabbat. The sounding of the shofar in this context is the one that marks the beginning of Shabbat when Yom Kippur is on Friday. Although there are several difficulties with this approach and the unfolding discussion does not seem to support it, this is the understanding adopted in the Jerusalem Talmud (see Tosafot and Rashba).
A second approach states that the Gemara is referring to a situation where Yom Kippur begins at the conclusion of Shabbat. According to these commentaries, the shofar would be sounded at that time, just as it was sounded every week at the conclusion of Shabbat, to divide between the sacred and the profane (Rambam).
Other commentaries explain that a special shofar blast was sounded on the eve of Yom Kippur, just like before Shabbat, and that is being discussed here.