If it is the day of ‘Ashoora’, and one of us ate, then after that found out that that day was the day of ‘Ashoora’, is it permissible for him to fast the rest of that day, based on the hadeeth which says: The caller of the Messenger of Allah (blessings and peace of Allah be upon him) called out on the day of ‘Ashoora’: “Whoever among you started the day fasting, let him complete his fast, and whoever has eaten, let him not eat for the rest of the day. Has anyone among you eaten today?” We said: Some of us have eaten and some of us have not eaten. He said: “Complete the rest of your day, both those who have eaten and those who have not eaten.” And they sent word to the people around Madinah, telling them to complete the rest of that day: “Let them fast the day of ‘Ashoora’, and whoever ate something in the first part of the day, let him fast the last part of it.”

Praise
be to Allah.

It is permissible for the one
who wants to observe a voluntary fast to form that intention during the day,
unlike the obligatory fast, for which it is stipulated that the intention
must be formed from the night before. 

In order for a voluntary fast to
be valid when the intention is formed during the day, it is stipulated that
before forming that intention there should not have been anything that is
contrary to fasting, such as eating, drinking and so on. If the person did
anything before forming the intention that invalidates the fast, then it is
not valid, and there is no difference of scholarly opinion concerning that.

End quote from al-Mulakhkhas
al-Fiqhi (1/393) 

Ibn Qudaamah al-Maqdisi said:
Once this is established, one of the conditions is that the person should
not have eaten before forming the intention, or have done anything that
invalidates the fast. If he did any of these things, then fasting is not
valid, and there is no difference of scholarly opinion concerning that as
far as we know.

End quote from al-Mughni
(3/115) 

With regard to the hadeeths
which speak of the Prophet (blessings and peace of Allah be upon him)
instructing the people to complete the fast of ‘Ashoora’, regardless of
whether or not they had eaten at the beginning of the day, the way in which
this may be understood is that the fast of ‘Ashoora’  was obligatory upon
them at that time. 

With regard to the obligatory
fast, it is obligatory for the one who finds out about it during the day to
stop eating and drinking from the time he finds out about it. 

Al-‘Ayni said concerning the
fast of ‘Ashoora’: It was an obligatory fast.

End quote from ‘Umdat al-Qaari’
(10/304) 

Al-Haafiz Ibn Hajar said: From
many hadeeths on this topic, it may be understood that it was obligatory (to
fast ‘Ashoora’), because the command to fast it is proven, then that was
confirmed, and then confirmed again by the general call, and by the
instruction to the one who had eaten to refrain from eating for the rest of
the day, then by the instruction to mothers not to breastfeed their children
during the day. Ibn Mas‘ood said: What is proven in Muslim is: When Ramadan
was enjoined, ‘Ashoora’ was abandoned, but it should be noted that
encouragement to fast it was not abandoned; rather that (encouragement)
remains in effect, which indicates that what was abandoned was the
obligation to fast it.

End quote from Fath al-Baari
(4/247) 

Imam an-Nawawi (may Allah have
mercy on him) said:

“Whoever did not fast, let him
fast, and whoever has eaten something, let him complete his fast until night
comes.” According to another version: “Whoever started the day fasting, let
him complete his fast, and whoever started his day not fasting, let him
complete the remainder of his day.” 

What these two reports mean is
that whoever intended to fast should complete his fast, and whoever did not
intend to fast but has not eaten anything, or he has eaten something, let
him refrain from eating for the remainder of that day, out of respect for
that day. This is like the situation where a person starts the day not
fasting when there is some doubt about the day, then it is confirmed that it
is indeed the first day of Ramadan. He must refrain from eating and so on
for the remainder of that day, out of respect for that day.

End quote from Sharh Saheeh
Muslim (8/13) 

Al-Baaji said: This is like the
case of the one who finds out that the day is Ramadan after part of the day
has passed; he has to refrain from eating regardless of whether he has
already eaten or not. 

End quote from al-Muntaqa
Sharh al-Muwatta’ (2/58)

 But after the fast of Ramadan
was enjoined, and ‘Ashoora’ became mustahabb (encouraged), this ruling no
longer applied to it; rather it came under the same ruling as all other
optional fasts: it is permissible to fast on that day with an intention
formed  on the day itself, on condition that one has not committed, prior to
that, any action that invalidates the fast. 

Shaykh Ibn ‘Uthaymeen was asked
about a person who did not realise that it was the day of ‘Ashoora’ until
partway through the day; is it valid for him to refrain from eating and
drinking for the remainder of that day, noting that he had eaten at the
beginning of the day? 

He (may Allah have mercy on him)
answered:

If he refrains from eating and
drinking for the remainder of that day, his fast is not valid, because he
ate at the beginning of the day. A voluntary fast is only valid from partway
through the day for the one who has not consumed anything that breaks the
fast from the beginning of the day. But if he has consumed anything that
breaks the fast at the beginning of the day, then his intention to fast by
refraining from eating and drinking for the remainder of the day is not
valid. Based on that, it will not benefit him to refrain from eating and
drinking so long as he has already eaten or drunk or done any of the things
that break the fast at the beginning of the day. 

End quote from Fataawa Noor
‘ala ad-Darb (11/2) 

And Allah knows best.