He started to observe a naafil (supererogatory) fast then he was invited to eat and he ate
If a Muslim intends to fast one day and starts to fast, then he wants to break that fast, he may do so, because completing a naafil fast is not obligatory. But it is mustahabb for him to complete it if he has no reason for not doing so or no reason for breaking it; if there is a reason for breaking his fast there is no sin in doing so. This is indicated by a number of ahaadeeth, including the following:
1 – Muslim (1154) narrated that ‘Aa’ishah (may Allaah be pleased with her) said: The Prophet (peace and blessings of Allaah be upon him) entered upon me one day and said: “Do you have anything (any food)?” We said, “No.” He said, “Then I am fasting.” Then he came to us another day and we said, “O Messenger of Allaah, we have been given some hays (a kind of food made from dates and ghee).” He said, “Show it to me, for I started the day fasting.” Then he ate.
This clearly supports the view of al-Shaafa’i and those who agreed with him, that a naafil fast may be broken, and one may eat during the day and cancel the fast, because it is naafil, and the person has the choice whether to start it and continue it. Among those who were of this view are a number of the Sahaabah, Ahmad, Ishaaq and others, but all of them, and al-Shaafa’i too, are agreed that it is mustahabb to complete the fast. Abu Haneefah and Maalik said that it is not permissible to break the fast and doing so is a sin. This is also the view of al-Hasan al-Basri, Makhool and al-Nakh’i; they said that it is obligatory for the one who breaks the fast without an excuse to make it up. Ibn ‘Abd al-Barr said: They were unanimously agreed that the one who has an excuse for breaking the fast does not have to make it up. And Allaah knows best.
2 – Ahmad (26353) narrated from Umm Haani’ that the Messenger of Allaah (peace and blessings of Allaah be upon him) entered upon her and called for something to drink, and drank some, then he passed it to her and she drank some. She said: “O Messenger of Allaah, I was fasting.” The Messenger of Allaah (peace and blessings of Allaah be upon him) said: “The one who is observing a voluntary fast is his own master; if he wishes he may (continue to) fast and if he wishes he may break his fast.” Classed as saheeh by al-Albaani in Saheeh al-Jaami’, 3854.
It says in Tuhfat al-Ahwadhi, commenting on this hadeeth:
The ahaadeeth on this topic indicate that it is permissible for the one who is observing a voluntary fast to break it, especially if he is invited to eat by a Muslim.
3 – al-Bayhaqi narrated that Abu Sa’eed al-Khudri said: Some food was made for the Prophet (peace and blessings of Allaah be upon him) and when it was served a man said: “I am fasting.” The Messenger of Allaah (peace and blessings of Allaah be upon him) said: “Your brother has invited you and make this effort for you. Break your fast and make it up later if you wish.” Al-Haafiz said: Its isnaad is hasan.