Half a Day of Yom Tov – Eruvin 39a-b
נוהגין אותו היום קודש ולמחר קדש
From the ruling that if the witnesses come late in the day, both that day and the next day are treated as Rosh HaShanah, R' Yose proves that both days are one continuous kedushah. The Sages, on the other hand, maintain that the first day is kept as a Yom Tov only so people will not come to make a mockery of Yom Tov.
Rabbi Elchanan Wasserman (Kovetz Shiurim to Beitzah 5a) finds the position of R' Yose difficult to understand: In the final analysis, surely R' Yose also concedes that the "first day" is not truly Rosh HaShanah. So, in the end, isn’t his position based on making a mockery of Yom Tov as well?
Reb Elchonon suggests that the difference of opinion hinges on the following question: Is Yom Tov (or Shabbos) a single united entity of continuous time or a composite entity of sequential time. R' Yose is of the opinion that a Yom Tov is one single unit. Hence, if one began observing a day as a potential Yom Tov (even as a rabbinic requirement), the rest of the day is, perforce, also Yom Tov. The Sages, on the other hand, are of the opinion that a Yom Tov is a composite of many "Yom Tov moments." Hence, were it not for the issue of making a mockery of Yom Tov, when the witnesses tarried, it would have been acceptable (even me'd'Rabbanan) to end Yom Tov in the middle of the day.
[This question is reminiscent of the well-known question as to whether the state of marriage is a single united entity of continuous time or a composite entity of sequential time. It is related that Rabbi Shimon Shkop once suggested to Rabbi Chaim Soloveitchik that marriage is a composite of many "Kiddushin moments," and that Reb Chaim responded: "If so, you deserve a Mazal Tov!]