What are you supposed to do if you inadvertently broke your fast*? Do you:
- keep going;
- extend it (if so, how long);
- cancel it (if so, do you make it up a different day)?
*This question is for day-fasts only.
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The Mishna Brura OC 549 sk 3 says to continue fasting if you accidentally ate.
In OC 568 sk 3 he says you can still say Aneinu at Mincha if it is a public fast day (as opposed to a personal one).
In OC 568 sk 8 he says that you do not need to fast again on a different day for accidental eating on the public fasts as well as any personal fast that has a fixed date (eg Yahrtzeit).
This halacha is not limited to Yom Kippur and Tish'a b'Av, but applies to "minor" fasts, as well.
From the Dirshu Daf Hayomi B'Halacha:
Must a person who mistakenly eats on a fast day continue fasting?
A person who mistakenly eats or drinks on either a ta’anis tzibur or a private fast that was accepted with a specific kabbalah (type #2 above) is obligated to continue fasting for the rest of the day. If, however, a person mistakenly ate or drink on a private fast that did not have a specific kabbalah (type #1), he does not need to continue fasting for the rest of the day. Nevertheless, he must fulfill his promise to fast on another day. A fast is only considered to be broken if a person ate at least a k’zayis of food or drank a m’lo lugmov (a cheek-full) of liquid. Some say that one must drink a revi’is to break the fast. According to Shulchon Aruch, if a person mistakenly breaks a private fast that only had a kabbalah during Mincha (type #3), he must continue fasting and does not need to fast another day. Rema, however, maintains that although he must continue fasting, he must also fast another day to make up for the broken fast.
[שו"ע תקסח, א, משנ"ב א, ב, ה, ז ו־ח, וביה"ל ד"ה ויש; ביאורים ומוספים דרשו, 2 ו־5]
How to do teshuva for mistakenly breaking a fast
As we have learned, a person who mistakenly eats on a public fast day must continue fasting until the end of the day. He is permitted to say Aneinu during his Shemoneh Esreh. He should replace the sentence of b’yom tzom ta’aniseinu with b’yom tzom ta’anis zeh, since he personally did not fast a whole day. It is not necessary to fast for an additional day, since the obligation to fast was pegged to a specific day. Nonetheless, some Rishonim say that as an act of teshuva (for mistakenly breaking his fast) he should fast on a different day. Others say that he should accept a Ta’anis Behab--a set of fasts for a Monday, Thursday and Monday. Some Achronim explain that fasting for three days is only necessary if a person ate throughout the fast day, not just once by mistake. If a person finds it difficult to fast, he can receive atonement by giving money to tzedakah.
[משנ"ב תקסח, ג ו־ח; ביאורים ומוספים דרשו, 1 ו־10]