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

If the people did not find out that the month had begun until partway through the day, should they stop eating and drinking? If they do stop eating and drinking, do they have to make up that day?
Praise be to Allah.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.