If they did not find out that the month of Ramadan had begun until partway through the day, what should they do?

If the
people did not find out that the month of Ramadan had begun until partway
through the day, what they must do is stop all things that invalidate the
fast for the rest of that day. 

The
evidence that it is obligatory to stop eating and drinking is as follows: 

~1~

The
words of Allah, may He be exalted (interpretation of the meaning):

“So
every one of you who is present (at his home) during that month should spend
it in fasting”

[al-Baqarah 2:185]. 

~2~

Al-Bukhaari (1900) and Muslim (1080) narrated that Ibn ‘Umar (may Allah be
pleased with him) said: I heard the Messenger of Allah (blessings and peace
of Allah be upon him) say: “When you see it then fast.” So the obligation of
fasting is connected to seeing the new moon. In this case the new moon has
been seen, so it is obligatory to fast. 

~3~

Al-Bukhaari (2007) narrated that Salamah ibn al-Akwa‘ (may Allah be pleased
with him) said: The Prophet (blessings and peace of Allah be upon him)
instructed a man of Aslam to call out to the people: “Whoever has eaten, let
him fast for the rest of the day, and whoever has not eaten, let him fast,
for today is the day of ‘Ashoora’.” 

As to
whether it is obligatory to make up that day, there is a difference of
opinion among the scholars. Some of the scholars – the majority – think that
it is obligatory to make up this day as well as refraining from eating and
drinking for the remainder of the day. They quoted as evidence for that the
report narrated by at-Tirmidhi (730) from Hafsah (may Allah be pleased with
her), that the Prophet (blessings and peace of Allah be upon him) said:
“Whoever did not intend to fast before dawn, his fast is not valid.” It was
classed as saheeh by al-Albaani in Saheeh Sunan at-Tirmidhi. They
said: In this case there was no intention to fast the night before, so the
fast is not valid. Refraining from eating and drinking for the rest of that
day is only obligatory out of respect for that time. 

Ibn
Qudaamah (may Allah have mercy on him) said:

If a
person started the day not fasting, believing that it was still Sha‘baan,
then proof was established that the new moon had been seen, he must refrain
from eating and drinking for the rest of the day and make up that day’s
fast, according to the majority of fuqaha’.

End
quote from al-Mughni (3/34). 

Shaykh
Mansoor al-Bahooti (may Allah have mercy on him) said:

“If
proof was established that the new moon had been seen” i.e., the new moon of
Ramadan

“during the day” i.e., the proof reaches them during the day

“then
they must” – this refers to those for whom fasting is obligatory

“refrain from eating and drinking, even if that is after having not fasted”
because it is not possible to refrain from eating and drinking for the
entire day, so they must do what they are able to do, because of the hadeeth
“If I instruct you to do something, do as much of it as you can.”

And
they have to “make up” that day, because it is proven to be a day of
Ramadan, but they did not fast it in the proper manner, so they have to make
it up.

End
quote from Kashshaaf al-Qinaa‘ (2/310) 

The
other scholarly view concerning this matter is that it is obligatory to
refrain from eating and drinking for the rest of that day, but not to make
it up. This is the view favoured by Shaykh al-Islam Ibn Taymiyah (may Allah
have mercy on him).

 This
view is supported by the hadeeth of Salamah ibn al-Akwa‘ quoted above about
the fast of ‘Ashoora’. There is no proof that those who ate at the beginning
of the day of ‘Ashoora’ made it up later on, despite the fact that fasting
‘Ashoora’ was obligatory at the beginning of Islam. 

It is
also supported by the fact that requiring people to refrain from eating and
drinking without it being a valid fast, in addition to the command to make
up that day, puts an extra burden on the individual with no evidence for
it. 

Shaykh
al-Islam Ibn Taymiyah (may Allah have mercy on him) said: If proof is
established during the day that the new moon of Ramadan has been seen, then
he should complete the rest of that day (refraining from eating and
drinking), and he does not have to make it up, even though he had eaten
(earlier in the day).

End
quote from al-Fataawa al-Kubra (5/376), 

Al-Mirdaawi (may Allah have mercy on him) said: Shaykh Taqiy ad-Deen said:
He should refrain from eating and drinking (for the rest of the day) and he
does not have to make it up. But if he did not find out that the new moon
had been seen until after sunset, he does not have to make up that day.

End
quote from al-Insaaf (3/283) 

Shaykh
Ibn ‘Uthaymeen (may Allah have mercy on him) said: With regard to the words
“If proof is established during the day, he must refrain (from eating and
drinking for the rest of that day) and making it up is required from
everyone for whom it was obligatory”, the word “proof” refers to proof that
the month had begun, either by sighting of the new moon or by completion of
thirty days of Sha‘baan. What is meant by “must refrain” is that he must
refrain from all things that invalidate the fast. 

This
is supported by the fact that when the Prophet (blessings and peace of Allah
be upon him) instructed the people to fast ‘Ashoora’ during the day, they
refrained from eating and drinking from that point. Moreover, it has been
proven that this day is part of Ramadan, so it is obligatory to refrain from
eating and drinking. 

With
regard to making it up, this means that it is obligatory to make up that day
during which proof was established that it was part of Ramadan. The reason
for that is that one of the conditions of an obligatory fast being valid is
that the intention should apply to the whole day, so it must come before
dawn, but the intention in this case came during the day, so they did not
fast a complete day. The Prophet (blessings and peace of Allah be upon him)
said: “Actions are but by intention, and each person will have but that
which he intended.” 

The
view that it is obligatory to make up the day in this case – i.e., when the
proof is established during the day – is the view of the majority of
scholars. 

Shaykh
al-Islam Ibn Taymiyah said: They have to refrain from eating and drinking
for the rest of the day, but they do not have to make it up. The reason for
that is that their eating and drinking before proof was established was
permissible, and Allah has permitted it to them, so they did not violate the
sanctity of the month. Rather they were unaware and they based their action
on what they thought was the case, which was that it was still Sha‘baan.
Thus they were included in the general meaning of the verse (interpretation
of the meaning): “Our Lord! Punish us
not if we forget or fall into error”
[al-Baqarah 2:286].
So they are like the one who ate thinking that it was still night, then he
found out that dawn had broken, or the one who ate thinking that the sun had
set, then he found out that it had not set. It is proven in Saheeh
al-Bukhaari from Asma’ bint Abi Bakr (may Allah be pleased with her) who
said: We broke the fast on a cloudy day at the time of the Prophet
(blessings and peace of Allah be upon him), then the sun appeared. And there
was no report that they were instructed to make it up. 

He
(may Allah have mercy on him) responded to the fact that they had not formed
the intention before dawn by noting that the intention comes after
knowledge, and they had no knowledge that the month had begun. If they had
no knowledge of something they could not do it, and Allah does not burden
any soul with more than it can bear. Therefore if they had delayed forming
the intention after finding out that the month had begun, their fast would
not be valid. 

His
argument and his answer are strong, but we do not feel quite comfortable
with his view and his analogy with the one who eats thinking that it is
still night or that the sun has set. That is subject to further debate,
because in this case the person had the intention of fasting, but he ate
thinking that it was still night or that it was daakhilan.

End
quote from ash-Sharh al-Mumti‘ (6/332-333) 

To sum
up:

refraining from eating and drinking is obligatory for the one who receives
news that the months has begun, even if that is during the day. With regard
to making up that day, there is a difference of opinion among the scholars
(may Allah have mercy on them). 

With
the development of means of communications in recent times, this issue has
become something that rarely arises. 

And
Allah knows best.