Taanit 12A

הא דאמרת מתענין לשעות ⁠— והוא שלא טעם כלום עד הערב. אמר ליה אביי: הא תענית מעלייתא היא! לא צריכא ⁠— דאימלך אימלוכי.

The halakha is that which you said, that one may fast for a few hours, provided that one took a fast of a few hours upon himself, and that he fasted and did not taste anything until the evening. Abaye said to Rav Ḥisda: This ruling is obvious, since it is a full-fledged fast, as one ultimately fasts the entire day. Rav Ḥisda answered: No, it is necessary to say this halakha in a case where he changed his mind, i.e., he began the day without intending to fast, but for various reasons he did not eat, and halfway through the day he decided to continue fasting for another few hours until nightfall. Rav Ḥisda maintains that this kind of fasting for hours is considered a fast.

ואמר רב חסדא: כל תענית שלא שקעה עליו חמה ⁠— לא שמיה תענית. מיתיבי: אנשי משמר מתענין ולא משלימין! התם, לצעורי נפשיה בעלמא הוא.

And Rav Ḥisda said: Any fast upon which the sun does not set, i.e., when one eats in the middle of the day, it is not called a fast at all. The Talmud raises an objection: The members of the priestly watch, the priests and Levites who are serving in the Temple that week, fast on a communal fast, like the members of the non-priestly watch who are attached to specific groups of priests, but they do not complete the fast with the rest of the community. This indicates that even a fast lasting only a few hours is called a fast. The Talmud explains: There they abstained from food merely to cause themselves distress, as an act of solidarity with the rest of the community, but this was not considered a full-fledged fast.